## A day at school

It was the English class in school. The teacher, a swarthy man with somewhat liberal political leanings, strode into the class: “Today we shall be studying a poem by the white American poet Robert Frost, Fire and Ice. Dandadipa, stand up and read the poem aloud.”
Dandadipa did as directed by the teacher:
Some say the world will end in fire,
Some say in ice.
From what I’ve tasted of desire
I hold with those who favor fire.
But if it had to perish twice,
I think I know enough of hate
To say that for destruction ice
Is also great
And would suffice.

The teacher then turned to Lootika and said: “Lootika, stand up and tell the class what you think about this poem you just heard.”
Lootika: “I think, if by world we mean the earth, undoubtedly, the end will come by fire. By fire I do not mean it in the real sense of a conflagration but more in the sense of intense heat.”
The teacher: “Why so young lady?”
Lootika: “The sun is an average yellow star and we know enough of astronomy today to accurately chart its course in time. It will evolve into an enormous distended star known as a red giant, about 100 or more times its current diameter. As this is happening all water on earth will evaporate and there will be lakes of molten aluminum and copper. Then heat of the sun will eventually bring the earth to an end by either vaporizing it completely or leaving behind just a little ball of iron.”
The teacher: “Lootika, that is very apocalyptic. I now see why the some of your science teachers say you are a special girl, while the rest declare you as a showoff.”
Lootika: “I am grateful to those who hold the former opinion but all this is common knowledge, sir.”

Tumul: “Sir, I think Lootika is simply showing off by talking nonsense. This poem is not about science as she tried to make it appear. Robert Frost is presenting two metaphors. In the first one he says that desires of people would lead to the end of humankind. That is what he means by the world ending in fire, where fire is the metaphor for all the conflicts arising from desires. When he says ice, he means the all the hatred that exists in the world. As we can see from events around us that could also end humankind.”

Having delivered his explanation, Tumul proudly surveyed his class, taking a glance at Lootika with the corner of his eye. The teacher almost sensing that Lootika might strike back raised his voice and said:
“Alright girls and boys. Keep quite and listen carefully. Tumul’s explanation about the metaphors is what the poet seeks to convey. However, let me tell you some history about this poem. The poet had met the astronomer Harlow Shapley. He told the Frost about the possible end of the earth stating that either the sun would explode and burn the earth or the earth would freeze in deep space. I believe what Lootika told us is something along the lines of the astronomer’s first explanation. Hence, while you are right Tumul, I don’t think you should be casting aspersion at Lootika’s literal explanation for something astronomical was the apparently inspiration for the poet’s opening lines about the end of the world in fire or ice. In this regard I would advise the class to watch the great serial presenting cosmology for the layman on Sundays by the black American scientist Floyd Mayweather.”

Somakhya was folding an origami model for their classmate Bobon of dark grayish yellow complexion from an erstwhile tribal category, who was a connoisseur of his artwork. The teacher turned to him and said: “Somakhya! This is not the craft class: pay attention!”
Somakhya stood up and said: “Sir it is just my hands; my mind and ears are all into your lesson on Fire and Ice.”
The teacher: “Then tell me what you think of this poem?”
Somakhya: “Lootika is correct insofar as the ultimate fiery end of the earth is concerned. There is also the snowball earth hypothesis, for which support is limited, that the earth froze over more than once. However, there are more distant icy scenarios, which we understand better since the days of Harlow Shapley. First, the sun after distending in to red giant, just as Lootika said, would eventually blow off its outer layers as a planetary nebula and be reduced to a hot small star called a white dwarf. Since the Sun is not a binary star, that white dwarf might in principle cool endlessly to reach a minimally thermal state called a black dwarf, which is a few degrees above absolute zero at some very, very distant time into the future. In one sense we could call this fate an icy end, like the second end the poet talks about.”

The teacher: “Left to yourself, you and Lootika would turn this into a science class. That said, as a rational person, I am happy that today we can derive our metaphors from science rather than religion. That is what people did in the past and Robert Frost might have also been influenced by the superstitious imagery of hell, which was described in western religious literature. Now, Jukuta, stand up and elaborate on the metaphorical aspects of the poem.”

Jukuta: “I think when he talks about desire he is referring to all the passions, like greed, gluttony and lust, which fuel anger and lead to a never-ceasing cycle of violence. The desire for territory, illicit drugs, taking other peoples’ wealth and the patriarchal oppression women leads to more warfare and oppression, which can only be compared to a conflagration. The lack of inter-personal amity and empathy is what he is paraphrasing as ice. We use the metaphor ‘cold’ to describe people with such qualities and it is root of hatred in this world. We see much of that too in this world and I am afraid it will be the cause for our end.”

The teacher: “That is excellent Jukuta. Do you all get the gist of this poem?”
Most of the class answered back in a chorus: “Yes”.
The teacher: “Somakhya I catch you roll your eyes as though all this is a joke. Stand up and explain clearly if you get the metaphorical point or not.”
Somakhya: “I hear what has been said but I think the poet lived in a different space and time. Methinks his diagnosis of the fire and ice were hence superficial.”
The teacher: “That is very arrogant on your part. I would rather see you back those flippant comments with more of an explanation.”
Somakhya: “What were listed by the other student as desires, namely lust, gluttony and the like, stem from basic biological imperatives. Long before Frost the buddha had also diagnosed these as ills, whose removal will bring us closer to a state of upliftment. I would say that for a living organism the elimination of biological imperatives cannot lead to a better state nor will they cause the end of the organism. We will rather see natural processes cause a convergence to a stable state. As for hatred, again, I do not see an end to biologically driven human conflict. If there were no other complicating factors, we will continue to see an arms race or some state of stabilization but not complete destruction. Rather, I would say that a pathological form of conflict engendered by the Abrahamistic religions is what might bring us to the brink of destruction.”

The teacher: “Somakhya I appreciate your articulation but I think this precisely the reason we need to study poetry. Your worldview seems quite set even at this young age and is inordinately naturalistic. We cannot be human with such a worldview. It is things like viewing matters in a more empathetic manner that will help you find your humanity. I should also admonish you for bringing in specific religions. All religions cause conflict. You are only showing prejudice by singling those which are not yours. Ours is a secular country and you should show the same respect you have for your religion to the religions of others.”

Lootika: “But sir history illustrates that all religions are not the same and some lie at the root of truly genocidal conflict.”
The teacher: “Alright, Lootika, don’t answer out of turn and keep that discussion for your history class. Students, we have had a good discussion. Now read the questions at the end of the poem and raise your hands if you do not understand them or have and doubts regarding the poem.”

Hemaling raised his hand: “Sir, I do not understand what metaphor means.”
Gomay raised his hand: “Sir, the first question: ‘what is the figure of speech in the opening lines of the poem?’ How to answer that one?”

The teacher busied himself answering these and other questions till the bell rang and the class came to a close for the recess. As the students spilled out of the room to head for lunch and take a break from the monotony of the classes, Lootika caught Somakhya’s eye and they exchanged a hand gesture. She shouted out to him: “We didn’t even get started with the aliens.” Somakhya: “May be that would have been less traumatic than my mention of ekarākṣasavāda!”

Then they moved on with their respective groups. Somakhya headed to a secluded spot for lunch along with Vidrum and Sharvamanyu. Sharvamanyu noted that Lootika and her gang similarly ensconced themselves in another secluded spot away from all other schoolmates: “That new girl seems like our Somakhya in more than one way. She seems to seclude herself while at lunch with her little gang which seems to be gathered from across different grades.”
Vidrum: “Those are her sisters and she is brāhmaṇa just like our guy”.
Sharvamanyu: “But then we are not brāhmaṇa-s. I believe Somakhya’s reasons are little more complex than that. He was quite suspicious I recall years ago before letting us dine with him.”
Somakhya: “I have told you’ll that before. It is more in the spirit of being a brāhmaṇa than the word of bhojyaprakaraṇa. The alien smells of meat, eggs and other items we perceive as being unclean, abhojya, are something that do not go well with the act of eating, which is one of sacrificing to the god Vaiśvānara within you. Also, the majority of our schoolmates do not understand the concept that we do not stand the act of eating from each other’s plates, given that they so indiscriminately share food, or drinking from others water bottles.”

Vidrum: “Yes. Yes. But I heard that you and Lootika will apparently be given a prize for your submission to the science essay contest on the evolution of human olfactory interest in Maillard reaction products. Does this have something to do with your brahminical sensibilities?”
Sharvamanyu: “First, what is this Maillard’s reaction?”
Somakhya: “It is a reaction in which the carbonyl groups, mostly from sugars, react with the amino groups coming from amino acids among other compounds and the oxygen and hydrogens go away as water leaving a compound with a C-N bond. It happens when food is heated a specific temperatures and the Maillard products are odoriferous imparting many of the favored flavors that are sought after in food. Our basic thesis was that human olfactory interest in Maillard products primarily arose after the use of fire became prevalent. The use of fire to prepare food we posit offered an advantage in making humans more likely to escape food-borne infection. Hence, we posited that those who evolved a liking for Maillard products were likely have a survival advantage due to that conferred by fire-cooked food.”

Sharvamanyu: “That is interesting. Somakhya, I have never seen you too interested in girls before. But I sense you have something for that new pretty girl?” Somakhya ignored his companion’s question and kept focusing on finishing his cold meal but Vidrum waded in: “Well, you may not know it but that is true. If she were to come here I am sure he would drop us and the lunch and they would lost in conversation on all manner of arcana.”
Sharvamanyu: “So may be that is why he said desire is not bad after all?”
Somakhya: “Desires may come and go but there are things deeper than that.”

Vidrum: “OK, OK. You know we do not read all this sciency stuff like you guys do. Nor do we watch Floyd Mayweather as the teacher asked us to do. He is so boring with his cliches of everything wanting to kill you.”
Sharvamanyu: “Still worse is his characterization of the reproductive system as an entertainment center in a latrine. We instead learn most of our stuff from movies. You said world might freeze into a snowball – saw movie like that. Could it really happen?”
Somakhya: “ That is entirely unclear. There are some lines of evidence that ice used to form in the tropics or even close to the equator at two points in the past history of the earth. But the evidence that it resulted in extinction of life is negative.”
Vidrum: “Just as we were coming out you guys seemed to suggest that aliens might bring our end? We saw a movie last weekend which showed something like that.”
Sharvamanyu: “Do you seriously think that would happen?”
Somakhya: “A very low probability event. In the 3 or more billion years of life on earth we don’t see evidence for single alien visitation beyond to two initial visitations that seeded the world with bacteria and archaea. But it if were to happen it could be catastrophic like for the animals of continents which had not encountered humans before. Or for that matter like the fate of the humans like the first Australians and Americans.”

The bell just rang then and they had to rush to back to class. That evening as Vidrum was nearing his home on the way back from school he saw Lootika and her sister Vrishchika hanging out at the corner of the street leaning on their bikes. Vidrum approached them : “So it seems you and Somakhya think that the end of the world could come from aliens? I just saw a movie last weekend which had such a plot.”
Lootika: “Possible but not very probable. Unlike what our teacher would want us to think, the end of our civilization could come sooner due to the unmatta-s. So I don’t think we need to keep an eye for those spaceships.”
Vrishchika: “And if it does happen I would place my bet not on a little green man or a furry abomination of an alien but a contagion like a virus or perhaps less likely some single-celled prokaryotic form.”
Vidrum: “So we should be guarding against those?”
Lootika: “Somakhya and I have been pondering about that of late. Especially on lines of why bacteria cause so many diseases but archaea hardly any. So at this stage we still seem to think this end, while possible, is certainly less likely than the fiery crash of an asteroid or a comet from space.”

Posted in Life |

## Mongolica: Qubilai Khan’s campaign to destroy the Southern Song

The final act in Mongol conquest of China shows the military capability of Qubilai and why his grandfather, the great Khan had singled him out as the one who someday would adorn his throne. We shall place here a very brief account of this war. In late summer of 1259 CE the Mongols faced a major crisis when the supreme leader Möngke Khan died of dysentery while conducting the siege of the fort of Diaoyu of the southern Song. The next in line for the throne of Chingiz was his brother Qubilai who was supported by his younger brother the Il-Khan Hülegü. Hülegü decided to send about half his men from Iran to aid Qubilai continue the campaign against the chīna-s. However, he was thwarted in this plan due to a squabble with his cousins of the Jochid line, who were in state of ferment after the death of their great leader Batu, the grandson of Chingiz via Jochi. Nevertheless, Hülegü manage to send the young Bayan of the Baarin tribe (one of the early tribes that elected Temüjin as Khan), a rising general in his army, to help his brother Qubilai. He was given a command at a young age after his father fell in battle while taking the Hashishin forts under Hülegü. Qubilai pressed on and established a bridgehead south of the Yangtze river near Ezhou. But soon thereafter he had to move back north due to his brother Ariq Böke claiming to be the great Khan. In the mean time the Song under their leader Jia Sidao regained the ground lost to the Mongols south of the Yangtze.

In 1260 CE after Qubilai Khan had settled the rebellion of his brother Ariq Böke, set his eyes on destroying the last Chinese kingdom, the Southern Song, once and for all. The Song were short on horses and thus lacked a swift-moving cavalry. However, Qubilai from his prior experience knew that the terrain meant that Song could still nullify the traditional cavalry tactics of the Mongols. Hence, he decided not to hurry and made elaborate preparations. The Mongols amassed extensive pyrotechnics as well also a powerful array of rockets and explosive bombs for sieges. But then Song had the largest army in the world of over a one million men and also had an impressive array of torsional artillery to deliver fireworks against a besieging force: large trebuchets with good accuracy and less accurate mangonels and also giant crossbows. Moreover, they were the first Chinese kingdom to have a permanent and strong navy fully capable of diverse maneuvers on rivers and the seas. It had an array of some of best ships of the time equipped with naval pyrotechnics for attacking coasts and other ships. They also had strategic depth in the form of the alliance with the kingdom of Campāvati to the south.

In 1265 CE Qubilai tested the Song land forces at Sichuan and comprehensively defeated them. This gave impetus to the Mongol morale to enter into a decisive war with the cīna-s. However, Qubilai knew fully well that this was not enough as Song from then on were going play a defensive strategy banking on their vast army. They were going block the key invasion routes that the Mongols would be able to take with a string of well-stocked forts. Thus, the Mongols would be forced to take routes through difficult terrain in central and south China or risk a completely new form of warfare, naval, against a force of over 100,000 Song marines who were stationed along the coast and instructed to preemptively to thwart any Mongol attempt at building a navy. Qubilai decided that he would first focus on the land campaign and then build up his navy over a period so that they could match up with the Song. He also raised a massive stock of pyrotechnics and bombs specifically designed to attack ships with the Song navy in eye.

Thus, in 1267 CE Qubilai directed his army to systematically proceed along the Han river, the left tributary of the Yangtze in central China with the Song capital of Hangzhou as the focus , even if the progress was slow. Putting this plan into action, the 30 year old general Bayan led the Mongols to a series of victories using heavy bombardment with the pyrotechnics and cutting off supplies with the mobile cavalry units, thus taking all the smaller forts which could be taken with just the land army. Then they besieged the second strongest Song fort of Fancheng and after a prolonged siege captured it. Provisioning the army with this captured fort, they advanced against the strongest and best defended Song fort Xiangyang in 1273. In the meantime Qubilai had readied the phase-I of the Mongol navy, put it on the river and carried out a continuous naval blockade of Xiangyang. Finally, with the land army converging, this mighty fort was captured by 1274. With that the Mongols broke into the core Yangtze region and closed in on Hangzhou. Qubilai promoted general Bayan for his successes as the commander-in-chief of the Mongol army. The Song lord Jia Sidao charged at the Mongols stationed at the Yangtze with a force of 130,000 but Bayan smashed his advance and forced him to retreat. He tried to negotiate a peace treaty but Bayan rebuffed it and continued the attack forcing Jia to flee. Then the Mongol generals Bayan and Aju punched their way forward aiming at Nanking, Changzhou and Wuxi. In the first city the Song army fled at the approach of the Mongols without offering much of a fight. In the subsequent encounters the Song lost heavily against the quick moving Mongol forces and had to surrender the cities. Several top Song generals were targeted and killed by Bayan and Aju in these battles thus denuding their command capacity.

Qubilai then sent three Chinese emissaries to discuss surrender terms with the Song but the Song killed those emissaries. Qubilai immediately ordered strikes on the city of Yangzhou on the Yangtze and Jiading and destroyed the Song units which were positioned there. He then used these as a base to launch a surprise attack with fast-moving boats equipped to hurl bombs and pyrotechnics on the first Song river fleet. The Song admiral taken by surprise was killed and his fleet was rapidly brunt and sunk. The veteran Song lord Jia Sidao was killed shortly thereafter by a fellow Chinese perhaps employed by the Mongols. Then the Mongols swept up the Jiangsu province where the Song population resisted strongly but was massacred upon being defeated. Hunan and Jingxi were taken next by Bayan. Then Qubilai launched a three pronged amphibious assault on the Song. A western wing under general Ajirghan marched to take the fort of Jiankang and secure the key Dusong pass. In the east Qubilai unfolded the second phase of his navy under admirals Dong and Zhang Hongfan to sail along the Yangtze to reach the sea and then secure the coast for a naval showdown with the main Song sea fleet. Bayan led the central wing to march straight to the capital.

Seeing the rapidly unfolding of the Mongol plans, and being reduced to a patch around Hangzhou, the Song sent a emissary stating that they were willing to be a protectorate under the Mongols. Bayan sent him back saying that the Mongols were now aiming for complete conquest of the Song. Finally, in March 1275 the Mongols closed in on the Song capital and launched a simultaneously attack with two land armies and one naval force. The Song thought they would stave off the Mongols with their fire-arrow giant crossbows as they had done to the elephant corps of the Han several centuries before. However, they came up against an overwhelming bombardment by the Mongols with thousands upon thousands of iron-cased and earthen bombs hurled from trebuchets. The Song crossbows and seige engines caught fire and parts of the capital province were were hit from the bay by the naval attack of Mongols. The superior pyrotechnics of the Mongol destroyed the Song naval defenses and allowed their ships to close in on the capital. Then the Mongols launched their main land assault with the cavalry division under Bayan. The Song while having a larger force numerically could not match up to the tulughama-like sorties of the Mongols and eventually folded up. With that the main Song land resistance was over and the Song queen surrendered the capital to the Mongols without any resistance in February of 1276 CE.

There was still a mighty Song fleet and the loyalists taking the two young surviving princes with them sailed down the sea and used Macau and nearby islands as a base. When Qubilai had to move north to face Qaidu who was challenging him as rival Khan the Song tried to reestablish themselves by fomenting rebellion in Fujian, Guangdong and Guangxi in 1277 and 1278 CE. Qubilai having settled the issue of Qaidu for the time being returned to finish the Song rebellion. The Mongols first flushed them out of Fujian and Guangxi by repeated land attacks and then corralled them in Guangdong where one of the Song princes died leaving the last one as the emperor. Seeing the Mongols close in, he and his supporters realized that there was no hope of fighting a land battle. But their navy of nearly 1000 excellent ships and several hundred supporting boats was intact and they decided to retreat to the island of Yaishan off Macau. It was here that the final battle was fought in 1279. The Song arrayed their ships in a rectangular formation and placed several palisades tied to boats to form a perimeter. Thus, the whole array was like a floating fortress from behind which the Song troops could fight. They also kept close to land so that they could supply their men with weapons, food and material for repair. This effectively cost their mobility against a Mongol fleet with smaller but much faster vessels. At first the Mongols took some high positions facing the Yaishan coast and launched a bombardment in February of 1279 with incendiary shells and stones. This damaged several ships of the Song and demoralized them to a degree.

However, when the Mongol fleet finally assembled for battle in March 1279, it was still only half the size of the Song fleet. On March 19th the Mongols calculated that the tide would provide two opportunities that day. The tide receded towards the south early in the morning creating a rush between the two islands where the Song fleet was stationed. Since they had built highly maneuverable ships, the Mongols used the momentum of the tides to launch rapid attack on the Song from the north. For this assault the Mongols chose the admiral Li Heng, said to be a descendant of the old emperor Tai-zong of the Tang. This attack brought the Mongols close to the Song ships and they began hand to hand fighting at which they excelled with the corps on the support boats. Taking advantage of this encounter the Mongols moved a second formation of about 100 ships to the south and launched an attack by mid-morning from the south. The Song turned the full force of their naval artillery against this formation. But by noon the Mongols made an amphibious landing on the island of Yaishan and made a pyrotechnic attack against the Song ships closest to land. By then the fourth Mongol fleet of around 100 completed the encirclement of the large but immobile Song fleet. Shortly after noon the Mongols breached the Song palisade and were now able to attack ships inside it. They then used the momentum of the afternoon tides to launch another rapid attack to close in on the Song before they could deploy their next round of naval artillery. In this attack they broke up the defensive rectangle of the Song completely and got to the main ships. Having boarded them they engaged in fierce closen combat. By evening the Mongols had killed 100,000 Song marines and the Song prince’s corpse was seen floating in the sea. By then Mongols had captured 800 Song ships. Seeing this, the Song navy lost heart and surrendered. With that the Song empire of China came to a close and the survivors fled to Campāvati. This closing battle also showed how far the Mongols had come from a horse-borne steppe power to one which could defeat one of the best navies of the time at sea.

In my childhood, one nice afternoon, I leaned against the cot in my room and lapsed into a reverie. This war flashed in before my eyes in great detail like a movie. Inspired by that we staged an enactment of the same which gave us much pleasure.

Posted in History |

## The quotient triangle, the parabola-hyperbola sequence, the remainder triangle and perfect numbers

The quotient triangle
Consider a positive integer $n$. Then for all $k=1,2,3...n$ do the floor operation $T_q[n]= \left\lfloor \tfrac{n}{k}\right\rfloor$. Say $n=10$, we get $T_q[10]=10, 5, 3, 2, 2, 1, 1, 1, 1, 1$, a sequence of quotients of the division $n \div k$. If we do this for all $n=1, 2, 3, 4 ...$ we get the quotient triangular array $T_q[n,k]$ whose top few elements are show below.

$\begin{tabular}{|*{10}{r|}} \cline{1-1} 1\\ \cline{1-2} 2 & 1\\ \cline{1-3} 3 & 1 & 1\\ \cline{1-4} 4 & 2 & 1 & 1\\ \cline{1-5} 5 & 2 & 1 & 1 & 1\\ \cline{1-6} 6 & 3 & 2 & 1 & 1 & 1\\ \cline{1-7} 7 & 3 & 2 & 1 & 1 & 1 & 1\\ \cline{1-8} 8 & 4 & 2 & 2 & 1 & 1 & 1 & 1\\ \cline{1-9} 9 & 4 & 3 & 2 & 1 & 1 & 1 & 1 & 1\\ \cline{1-10} 10 & 5 & 3 & 2 & 2 & 1 & 1 & 1 & 1 & 1\\ \cline{1-10} \end{tabular}$

It is obvious that the first column $T_q[n,1]$ is the sequence of $n$ itself. The second column $T_q[n,2]$ is the sequence of positive integers, each repeated twice; the third column $T_q[n,3]$ is the sequence of positive integers, each repeated thrice; so on.

It we linearize this triangular array we get the sequence $f_q= 1, 2, 1, 3, 1, 1, 4, 2, 1, 1...$. Since each row of the triangular array adds $n=1, 2, 3, 4...$ elements to the sequence, it grows as the sum of numbers from 1:n. We see that the successive maxima are attained at the following terms of $f_q$: 1, 2, 4, 7, 11, 16, 22, 29, 37, 46, 56… We can plot $f_q$ by placing the first term $f_q[1]$ at 0, i.e., its x-coordinate will be 0, while its y-coordinate will be $f_q[1]$; for $x=1, y=f_q[2]$ and so on. Based on the sum of integers 1:n we can show that the successive maxima attained by the sequence would be bounded in this plot by the parabola (Figure 1):

$y=\dfrac{\sqrt{1+8x}-1}{2}+1$

Figure 1

This sequence $f_q$ also features cycles of decaying values (Figure 1) between successive maxima, starting from one maximum and going to 1 before jumping back to the next maximum and decaying again. Since the quotients are successively determined by $n/1, n/2, n/3...3 = n\big/ n/3, 2 = n\big/ n/2, 1 = n\big/ 1/n$, each cycle of the sequence of length $n$ is a discrete form of the rectangular hyperbola (Figure 2):

$y=\dfrac{n}{x}$

Figure 2

Thus, we may term sequence $f_q$ the parabola-hyperbola sequence. Since any $n/2 < k \le n$ will produce quotient 1, the fraction of 1s in $f_q$ will converge to $f(1)=1-1/2=1/2$; similarly, $n/3 < k \le n/2$ with produce quotient 2; hence, the fraction of 2s in $f_q$ will converge to $f(2)=1/2-1/3=1/6$. Thus, generally $f(k) \sim \tfrac{1}{k(k+1)}$.

We then define a summatory sequence $\varsigma[n]$ such that each term is the sum of the nth row of the quotient triangular array $T_q$:

$\varsigma[n]=\displaystyle \sum_{k=1}^n T_q[n,k]$

This sequence goes thus: 1, 3, 5, 8, 10, 14, 16, 20, 23, 27… and is shown graphically in figure 3.

Figure 3

Each term of this sequence is the sum of the terms in a given hyperbolic cycle (Figure 2) of sequence $f_q$ (Figure 1). Thus, it can be approximated in continuous form by the area under the hyperbola corresponding to each cycle $y=n/x$. Hence, we can create a continuous approximation for $\varsigma[n]$ by the integral function:

$y=\displaystyle \int_1^x \dfrac{x dt}{t}=x\log(x)$

This is shown by the red curve in Figure 3. We notice that this continuous approximation falls short of actual discrete $\varsigma[n]$ sequence. This correction factor can be empirically determined using the method of least squares to be a linear term. Thus, with this correction the below function provides a good fit for the average behavior of $\varsigma[n]$:

$y=x\log(x)+0.154 x$

The empirical determination of this correction term prompted us to try to determine it from first principles. The function $y=x\log(x)$ is a continuous approximation of the area in one cycle of the sequence $f_q$. But in reality our $f_q$ cycles are approximately the discrete harmonic series as opposed to a smooth hyperbola. From Euler’s work we know that the difference between the discrete form and the continuous integral converges to that mysterious number Euler’s constant $\gamma= 0.5772157...$ for a unit hyperbola. To make use of this we tried some experiments with the hyperbola and realized that the best way to capture the area of the cycle was to use the symmetry of the hyperbola $y=n/x$ about the line $y=x$ (Figure 2). Here, we can count all $n/k$ in a cycle vertically till $\sqrt{n}$ (see vertical line in Figure 2). The due to symmetry we can make the count again from the other end horizontally till $\sqrt{n}$. This way we would have completely covered all the discrete counts except that we would have counted the square $\sqrt{n} \times \sqrt{n}$ twice. Hence, need to subtract $n$ from our sum. We can get the two symmetric discrete sums now by the addition of $\gamma$ to the integral of the continuous form. Thus, we can write the approximate area for a cycle and hence the function approximating $\varsigma[n]$ as:

$y=x(2\displaystyle \int_1^{\sqrt{x}} \dfrac{dt}{t}+\gamma)-(\sqrt{x})^2= 2x(\log(\sqrt{x})+\gamma)-x=2x(\dfrac{1}{2}\log(x)+\gamma)-x=x\log(x)+(2\gamma-1)x$

Thus, we get our correction term to be $2\gamma-1=0.1544313...$. It gave us great pleasure to have figured this out from scratch without any study of mathematical literature in this regard. We noticed that this already good fit can be made even better by addition of a further constant term $c=5.479$. We do not know if this is really a constant or is some subtle term currently beyond our reach. Thus, we may write the final function approximating summatory $\varsigma[n]$ sequence as:

$y=x\log(x)+(2\gamma -1)x+5.479$

This function gives a mean difference of $6.8 \times 10^{-4}$ with $\varsigma[n]$ for $n=1:30000$ (Figure 3: light blue dashed line). While this captures the average behavior of $\varsigma[n]$, examination of the specific behavior of $\varsigma[n]$ shows that it exhibits saltations that are greater than usual at certain values. To better understand this we created the difference sequence $\Delta \varsigma[n]$ (Figure 4):

$\Delta \varsigma[n]= \varsigma[n+1]-\varsigma[n]$

Figure 4. We are yet to figure a curve to fit the successive maxima attained by $\Delta \varsigma[n]$.

The sequence $\Delta \varsigma[n]$ shows several interesting features which we consider in detail below:
1) The lowest value it ever attains is 2. A closer examination of the indices at which $\Delta \varsigma[n]=2$ reveals that they correspond to $n+1$ being a prime in the parent $\varsigma[n]$ sequence. Thus, there are as many minima in $\Delta \varsigma[n]$ as there are primes. This can be explained thus: Since, a prime is completely divisible only by $k=1, n$, these two values will generate quotients of 1 and $n$ respectively to add to the quotient sum. The remaining quotients will be the same as previous number as none of the other $k$ between 1 and $n$ will perfectly divide the prime. Thus, the quotient sum will minimally differ from that of the previous number by 2.

2) Jumps above the median value of 6 (for this range; Figure 4: violet line) have propensity to increase with increasing $n$. Analysis of these jumps revealed that they have a significantly higher propensity to occur at $n=6k-1$. Figure 5 shows a box-plot of $\Delta \varsigma[n]$, which indicates that the median value of this sequence for $n=6k-1$ is significantly higher than median value of the sequence for any other $n$ as well as the median value of the overall sequence. This can be explained by considering the following $n=6k-1$ corresponds to $n=6k$ in the parent $\varsigma[n]$ sequence. A number of the form $6k$ will undergo complete divisions by at least 1, 2, 3, 6 out of the first 6 $k$. However, the number before it $6k-1$ will not be divisible by 2, 3 or their multiples. Since it cannot undergo complete division by them, its quotients will be less than the corresponding ones of $6k$ by 1. Thus there will be a jump in $\Delta \varsigma[n]$ corresponding to $n=6k$ in $\varsigma[n]$. As e.g. consider 12:
$\varsigma[10]= 10+5+3+2+2+1+1+1+1+1=27$
$\varsigma[11]= 11+5+3+2+2+1+1+1+1+1+1=29$
$\varsigma[12]= 12+6+4+3+2+2+1+1+1+1+1+1=35$;
$\varsigma[13]= 13+6+4+3+2+2+1+1+1+1+1+1+1=37$
Thus, $\varsigma[12]$ causes a spike in $\Delta \varsigma[11]=6$ due to completion of divisions by 2,3,4,6.

Figure 5.

3) Notably, the successive record values of $\Delta\varsigma[n]$ for $n<30000$ are attained at: 1, 3, 5, 11, 23, 35, 47, 59, 119, 179, 239, 359, 719, 839, 1259, 1679, 2519, 5039, 7559, 10079, 15119, 20159, 25199, 27719 (Figure 4). These values of $n$ correspond to 2, 4, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720 in $\varsigma[n]$. Remarkably, these latter numbers are the highly composite numbers first defined by Ramanujan in 1915 CE. They are numbers with more divisors than any integer smaller than them. We can see why these highly composite numbers spawn maxima in $\Delta\varsigma[n]$ — they will be maximally completely divided and thus yield the largest quotient sum for any number up to them. Thus, $\Delta\varsigma[n]$ presents them as antipodes to the primes. Further, Ramanujan defined a second more inclusive set of numbers, the largely composite numbers, which are numbers with more or the same number of divisors than any integer smaller than them. This more inclusive set spawns high-points in $\Delta\varsigma[n]$, which are not maxima but still stand out (Figure 4).

As an aside one may note that the above $n$ at which the $\Delta\varsigma[n]$ maxima occur are often primes: 15 out of the 24. Most of these are also in particular Sophie Germain or safe primes (primes of form where if $p$ is a prime and $2p+1$ is also a prime. Then the former is Sophie Germain prime and the latter a safe prime).

4) The successive record values of $\Delta\varsigma[n]$ are: 2, 3, 4, 6, 8, 9, 10, 12, 16, 18, 20, 24, 30, 32, 36, 40, 48, 60, 64, 72, 80, 84, 90, 96… It turns out this sequence (another recorded by Ramanujan) is the number of divisors of the corresponding highly composite numbers which spawn the $\Delta\varsigma[n]$ maximum. Thus, the successive maximum values of $\Delta\varsigma[n]$ are attained at $n$ one less than a highly composite number and is equal to the number of divisors of the HCN.

5) From the the above discussion of the specific cases of the maxima and minima it should be apparent that:

$\Delta\varsigma[n]=\tau[n]$ where $\tau[n]$ is the number of divisors including 1 and $n$ itself for $n \ge 2$.

The remainder triangle
The above triangular array $T_q$ can be converted to a second triangular array by the below transformation:

$T_r[n,k]=n-k \cdot T_q[n,k]$

This $T_r$ turns out to be the triangular array of remainders. This can be alternatively obtained thus: Consider a positive integer $n$. Then for all $k=1,2,3...n$, $T_r[n] \equiv n \mod k$. Say $n=10$, we get $T_r[10]=0, 0, 1, 2, 0, 4, 3, 2, 1, 0$, a sequence of remainders of the division $n \div k$. If we do this for all $n=1, 2, 3, 4 ...$ we get the remainder triangular array $T_r[n,k]$, which is the same as the array obtained by the above-stated transformation of $T_q[n,k]$. Shown below are its initial terms.

$\begin{tabular}{|*{10}{r|}} \cline{1-1} 0 \\ \cline{1-2} 0 & 0 \\ \cline{1-3} 0 & 1 & 0 \\ \cline{1-4} 0 & 0 & 1 & 0 \\ \cline{1-5} 0 & 1 & 2 & 1 & 0 \\ \cline{1-6} 0 & 0 & 0 & 2 & 1 & 0 \\ \cline{1-7} 0 & 1 & 1 & 3 & 2 & 1 & 0 \\ \cline{1-8} 0 & 0 & 2 & 0 & 3 & 2 & 1 & 0 \\ \cline{1-9} 0 & 1 & 0 & 1 & 4 & 3 & 2 & 1 & 0 \\ \cline{1-10} 0 & 0 & 1 & 2 & 0 & 4 & 3 & 2 & 1 & 0 \\ \cline{1-10} \end{tabular}$

We observe that each column in $T_r$ can be described thus: 1st, all 0s; 2nd a 2-term cycle sequence of 0,1; 3rd a 3-term cycle of sequence 0,1,2. In general terms the kth column is a $k$ term cycle of form $0,1,2...k-1$

If we linearize $T_r$ we get the remainders sequence $f_r$, which goes 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 2, 1, 0, 0, 0, 0, 2, 1… As the linearization of a triangular array $f_r$ also grows as $\sum_{j=1}^n j$, where $n$ is the row number of $T_r$. The sequence is plotted in Figure 6 with the x-coordinate of the first term as 0 and y=coordinate as $f_r[1]$.

Figure 6

It shows cycles of length $n=1,2,3..$. Each pair of cycles shows the same maximum value. Examination of the sequence shows that successively greater maxima are all bounded by the parabola:

$y=\dfrac{\sqrt{1+2x}-1}{2}$

Just as we did with $T_q$, we can similarly define the summatory sequence $\rho[n]$ (we are using this notation after we discovered post facto that $\rho$ has been used for related remainder sequences in the mathematical literature, where each term is the sum of the numbers in a row of $T_r$. This sequence goes thus: 0, 0, 1, 1, 4, 3, 8, 8, 12, 13, 22, 17, 28, 31, 36, 36, 51, 47, 64, 61… It is plotted in Figure 7.

Figure 7

We determined that the sequence can be described by a continuous function, a parabola of the form $y=a x^2$ (shown as red line in Figure 7), where we empirically determined $a \approx 0.177$.

We next sought to obtain the exact value of the constant of the parabola $a$ from first principles. For this consider the plot a single cycle of length $n$ of sequence $f_r$ (Figure 8). We observe that as $n \to \infty$ it is a series of right triangles.

Figure 8

The largest of these triangles is an isosceles triangle with $h=n/2, b=n/2$. The next triangle has $h=n/3, b=n/6$, the next has $h=n/4, b=n/12$. Thus, the general expression for the heights and bases of these triangles is $h=n\cdot\tfrac{1}{k+1}, b=n \cdot \tfrac{1}{k(k+1)}$ where $k=1,2,3...$ and the area of the triangles would be $A=n^2 \cdot \tfrac{1}{2k(k+1)^2}$. Thus at limit the area under the triangles in a single cycle, which will be the value of the nth term of $\rho[n]$ for large $n$, will be given by the sum (assuming $n \to \infty$):

$y=n^2\displaystyle \sum_{k=1}^\infty \dfrac{1}{2k(k+1)^2}$

We realized, much to our pleasure on discovering it, that this sum is convergent and remarkably can be expressed in terms of the Riemann $\zeta$ function as:

$\displaystyle \sum_{k=1}^\infty \dfrac{1}{2k(k+1)^2}= 1-\dfrac{\zeta(2)}{2}=1-\dfrac{\pi^2}{12} \approx 0.177532966$

Thus we can express the parabola fitting the sequence $\rho[n]$ precisely as:

$y=\left(1-\dfrac{\pi^2}{12}\right)x^2$

Thus, again we stumble upon one of those deep links we seen in mathematics, in this case between remainders of division and the number $\pi$. Now, to study the exact behavior of $\rho[n]$ we define the difference sequence $\Delta \rho[n]$:

$\Delta \rho[n] =\rho[n+1]-\rho[n]$

This difference sequence is plotted in Figure 9.

Figure 9. The points corresponding to perfect numbers and powers of 2 are shown in red and blue respectively.

The $\Delta \rho[n]$ sequence shows the following notable features:
1) $\Delta \rho[n]$ takes positive values more often than negative values (3.03:1) in the first 30000 terms. This corresponds to the ratio of numbers with the sum of their divisors smaller than themselves being less than them by at least 2 (known as deficient numbers) to those whose sum of divisors smaller than them is equal or greater than them. This will become clear from the discussion that follows further down.

2) Successive maxima, i.e. the record value of the sequence up to certain $n$ occur only at even values of $n > 1$. Closer examination revealed that the maxima are associated with $n$ being a prime in the parent sequence $\rho[n]$; then $\Delta \rho[n-1]$ will be a maximum. If we take a number $6k$ then the numbers $6k+2, 6k+4$ will be even. $6k+3$ will be divisible by 3. Thus, these will not be primes. That leaves $6k+1$ and $6k+5$; the latter is the same as $6k-1$ for another $k$. Thus, $6k \pm 1$ will not be divisible by 2 and 3 and could be prime $\ge 5$. Thus, all primes $p \ge 5$ can be expressed as $6k \pm 1$. Hence, in the sequence $\Delta \rho[n]$, $p=6k+1$ will correspond $n=6k$ and $p=6k-1$ to $n=6k-2$. Thus, at $n \ge 6$, the successive $\Delta \rho[n]$ maxima occur at $n=6k$ or $n=6k-2$ corresponding to every $n$ that related to a prime in the $\rho[n]$ sequence. That $\Delta \rho[n]$ successive maxima will occur at primes can be explained easily. A prime will not be divisible by any number other than 1 and itself so all other remainders will be non-zero. The number before it will have at least 2 or at least 2, 3 and 6 as divisors that will yield remainder 0. Thus, a prime will result in remainders adding to a larger number relative to the remainder sum of its prior number, thereby causing a maximum to attained. Thus, there will as many successive maxima in $\Delta \rho[n]$ as primes.

3) $\Delta \rho[n]=0$ only when $n=2^k-1$, which corresponds to $n=2^k$ in $\rho[n]$.

4) Negative values of $\Delta \rho[n]$ show a more complex behavior. They are seen where the $n$ in $\rho[n]$ is a number which might be perfect numbers, abundant numbers and highly abundant numbers. We discuss these in detail below.

5) The successive minima, i.e. record negative values assumed by $\Delta \rho[n]$ for a given $n$ are fewer than the successive maxima. While we see a succession of 3245 maxima, corresponding the same number of primes in the first 30000 terms of $\Delta \rho[n]$, we only see a succession of 60 minima. We did some experimentation to discover where these minima occur. Let us define an arithmetic function as below for all $n \ge 2$:

$s[n]=\left(\displaystyle \sum d_n[i]\right)-2n$. Where $d_n[i]$ is a divisor of $n$ from $d[i]=1...n$

Whenever $s[n]$ is greater than $s[k]$ of all $k$ integers lesser than it, then $n$ is considered part of the sequence. It goes: 2, 6, 12, 24, 36, 48, 60, 72, 84, 96… and may be called “remainder anti-primes”. When $n$ belongs to this sequence in $\rho[n]$ it spawns a minimum at $\Delta \rho[n-1]$. Thus, for $n\ge 5$ the successive minima occur only at a $n$ of the form $6k-1$. Remainder anti-primes are a subset of a class of numbers, highly abundant numbers, defined by the mathematician S. Sivasankaranarayana Pillai. These are numbers whose divisors smaller than themselves sum up to a value, which is bigger than the sum of the divisors for each of the numbers smaller than that number. More generally, whenever $n$ of $\rho[n]$ is a highly abundant number it results in strongly negative values in $\Delta \rho[n-1]$.

6) Definition of $s[n]$ also helps us understand the value assumed by $\Delta \rho[n]$:

$\Delta \rho[n-1]=-(s[n]+1)$

Thus, for a prime $p$ since the divisors are only $1, p$, we have $\Delta \rho[p-1]=-(p+1-2p+1)=p-2$. Thus successive maxima in $\Delta \rho[n]$ occur at $n=p-1$ and attain the value $p-2$. Thus, after 1 they are either of the form $6k-1$ or $6k-3$ and will always be odd. Further, this implies that the successive maxima are bounded by the line $y=x$ (Figure 9). The successive minima usually take the form $-6k \pm 1, -6k \pm 2, -6k \pm 3$. The middle of these, the only even values of minima, are very rare: there being on 3 in the first 30000 terms, namely $\Delta \rho[35, 71, 7199]= -20, -52, -10990$. The successive minima increase in magnitude with increasing $n$ but can attain much greater magnitudes than the maxima in their vicinity. Their increase with $n$ is not linear unlike the maxima as the $s[n]$ for this specialized subset of highly abundant numbers grows rapidly. We empirically found a curve of the form $y=-\tfrac{x\log(x)}{5.017}$ to approximately fit them (Figure 9).

7) When $\Delta\rho[n]=-1$ it corresponds to $n+1$, i.e. $n$ in $\rho[n]$ of 6, 28, 496, 8128 in our range of 1:30000. These numbers are known after the Pythagorean sage Nicomachus as perfect numbers because their divisors other than themselves add up exactly to them (one can see why this is so from the above, $\Delta \rho[n-1]=-(s[n]+1)$):
6=1+2+3
28=1+2+4+7+14
496=1+2+4+8+16+31+62+124+248
8128=1+2+4+8+16+32+64+127+254+508+1016+2032+4064
One notices that they have a curious form with respect to the powers of 2:
$6=2^0+2^1+2^2-2^0$
$28=2^0+2^1+2^2+2^3-2^0+2^4+2^1$
$496=2^0+2^1+2^2+2^3+2^4+2^5-2^0+2^6-2^1+2^7-2^2+2^8-2^3$
$8128=2^0+2^1+2^2+2^3+2^4+2^5+2^6+2^7-2^0+2^8-2^1+2^9-2^2+2^{10}-2^3+2^{11}-2^4+2^{12}-2^5$

More generally, if $p$ is a prime and $2p-1$ is also a prime (such primes are called Mersenne primes) then $2^{p-1}(2^p - 1)$ is a perfect number. The search for such perfect numbers goes on this date computationally and checking online while writing this article shows that to date 50 such numbers have been found. All known perfect numbers are of a form ending in 6 or 28. Yet it has not be formally proven that no odd perfect numbers exist or if there are an infinite number of perfect numbers!

8) Abundant numbers are those numbers whose divisors smaller than themselves add up to a number greater than themselves. e.g. $d(12)=1,2,3,4,6; 12<16$. When $n$ is an abundant number in $\rho[n]$ it results in a negative value of $\Delta\rho[n-1]$. The odd abundant numbers like 945, 1575, 2205 in $\rho[n]$ result in the only even indices ( $n-1$) at which the sequence $\Delta\rho[n]$ is negative.

9) $\Delta \rho[n]=-2$ is unknown to us. This would imply that the sum of the divisors of a number which are smaller than it can never be 1 more than the number. Similarly $\Delta \rho[n]=2$, appears to be unknown, i.e. corresponding to deficient numbers with a deficiency of 3. We do not know if these have been proven or disproven. $\Delta \rho[n]=1$ are also rare. We get only 3 such numbers from 1:30000 which correspond the below $n$ of $\rho[n]$ and observe that:
$3=3\times 1; \; 1=3-2$
$10=1\times 2 \times 5; \; 1+2+5=10-2$
$136=1 \times 2 \times 4 \times 8 \times 17 \times 34 \times 68; \; 1 + 2 + 4 + 8 + 17 + 34 +68 =136-2$
These are numbers whose sum of divisors smaller than them fall short of them by 2. Similarly, we have only 5 numbers resulting in $\Delta\rho[n]=-3$: 20, 104 464, 650, 1952. These are numbers whose sum of divisors smaller than them exceed them by 2.

Epilog
The $\Delta\varsigma[n]$ and $\Delta \rho[n]$ sequences (Figures 4 and 9) are striking in being simple means of illustrating the fundamental asymmetry in the number world between primes and their anti-numbers. $\Delta\varsigma[n]$ reveals the highly composite numbers, and more weakly the largely composite numbers of Ramanujan as the antipodes of primes. $\Delta \rho[n]$ suggests that the remainder anti-primes, a subset of the highly abundant numbers of Pillai, are the anti-numbers of primes. From the vantage point of these sequences the primes appear to define a more regular pattern in terms of minima and maxima respectively while these anti-numbers define more mysterious patterns.

There are some questions for which we do not have ready answers are (though they might have been answered/studied by mathematicians):
1) Can you give an exact form for the constant correction term used to recapitulating $\varsigma(x)$?
2) Can you derive a function that bounds the maxima of $\Delta\varsigma[n]$?
3) Can you derive the exact form of the bounding function of the minima of $\Delta\rho[n]$?

We carried out all this quite independently of the mathematical literature and were pleased to see the organic emergence of the two classes of anti-prime numbers including those previously discovered by Ramanujan. We showed an earlier variant of this material to a person with extraordinary mathematical talent. He brought to our attention a formal proof for the relationship between perfect numbers and the remainder sequence, which was given to him by a poorly known genius Suryanarayana from Vishakhapatnam who studied such issues extensively in the last century. However, such a proof might have been first published by the mathematician Lucas in the late 1800s. Searching the internet, we came across a simple presentation of that proof in an excellent paper by Spivey in the Mathematics Magazine in 2005. Hence, we are not providing that here. Otherwise our account here provides a summary of the main interesting results concerning these sequences we found as part of our investigation.

Posted in Scientific ramblings |

## Counting primes, arithmetic functions, Ramanujan and the like

We originally wished to have a tail-piece for our previous note that would describe more precisely the relationship between the Möbius function and the distribution of prime numbers. However, since that would have needed a bit of a detour in order to be clearer to an unfamiliar reader, we decided to tell that story more expansively and separately. All that will be said here is elementary stuff that has been narrated many times by many other people. Nevertheless, given that this is a mathematical story everyone likes to tell, we are also telling it in the way it impressed itself on our own consciousness.

What is the distribution of prime numbers? This is a question many of us might have wondered about in school. The typical school teacher, which we had, showed little interest in answering it. The number of prime numbers less than or equal to a number $x$ is termed $\pi(x)$ or the prime counting function. The 15 year old Carl Gauss spent his fallow time finding prime numbers in intervals of 1000s and thus built up an impressive list of them. From that he deduced an approximate expression for their distribution and wrote it down on his log tables:

$\pi(n) \sim \dfrac{n}{\log(n)}$

Here the $\sim$ notation means that as $n \to \infty$ the ratio of $\pi(n)$ to $\tfrac{n}{\log(n)} \to 1$, i.e., they are asymptotic. Later, his rival the French mathematician Legendre published a similar result:

$\pi(n) \sim \dfrac{n}{\log(n)-1.08366}$

But the Russian scientist Chebyshev showed that rather than 1.08366 the negative term would be correctly 1 as $n$ becomes large. Gauss subsequently improved his conjecture using the below function:

$\pi(n) \sim \textrm{Li}(n)$, where $\textrm{Li}(n)=\displaystyle \int_2^n \dfrac{dx}{\log(x)}$

His former student and professorial successor at Göttingen, Dirichlet, independently arrived at the same function, the logarithmic integral $\int_0^x\tfrac{dt}{\log(t)}$, as a possible expression for the asymptotic distribution of prime numbers, though he expressed it as a series. $\textrm{Li}(n)$ can be expressed as a series, which for large $n$, can give as close approximations of it as needed depending on number of terms we use. Such a series is termed an asymptotic series for $\textrm{Li}(n)$.

$\textrm{Li}(n) \sim \dfrac{n}{\log(n)} + \dfrac{n}{(\log(n))^2}+\dfrac{2n}{(\log(n))^3}+\dfrac{6n}{(\log(n))^4}...=\displaystyle \sum_{k=0}^\infty \dfrac{k!\cdot n}{(\log(n))^{k+1}}$

As a numerical example: $\textrm{Li}(10000) = 1246.137$. Using the first 4 terms of the above series we get a value of $1237.554$, which is a $.69\%$ error. $\textrm{Li}(100000) = 9629.809$. Again using the first 4 terms of the series we get $9605.549$, which is a a $.25\%$ error. One also notices that the first term of this series yields Gauss’s original cruder approximation. Figure 1 shows $\pi(x), \tfrac{x}{\log(x)}, \textrm{Li}(x)$. The proofs of these conjectures were first obtained in the later 1800s after Riemann’s discovery of the complete $\zeta$ function and came to be known as the famous prime number theorem.

Figure 1 $\pi(x)$ in red, $\tfrac{x}{\log(x)}$ in blue and $\textrm{Li}(x)$ in green.

Now, this function $\textrm{Li}(x)$ was already discovered before Gauss by Leonhard Euler but its first extensive study is part of a story of the lives and times of men. When a supernova like Gauss shines in the firmament the light of other stars, even if bright, gets drowned. That indeed was the fate of his compatriot Johann von Soldner. Born about an year before Gauss, von Soldner was another of those self-taught geniuses who at an early age showed a talent both in mathematics and making instruments of his own. He went on to be scientist of note and a land surveyor (even as Gauss). Among other things, he was one of the first to predict the bending of light by gravity (of course Newton himself believed that gravity would act on light particles and bend their path but von Soldner made numerically precise predictions of what the value would be). He proposed an experiment of observing stars in the near the Sun in the sky to test this proposed bending (something later measured by the famous faked experiments of Arthur Eddington as part of testing Einstein’s prediction of the same from his theory of relativity). In 1809 CE, von Soldner using his great capacity for numerical calculations studied $\textrm{Li}(x)$ and provided a table of its values up to 7 decimal places. In the process he also calculated Euler’s constant $\gamma=0.577215...$ that plays a key role in evaluating this integral to more than thirty places. He obtained a different value from the 19th place onwards for $\gamma$ than that obtained by earlier by geometer Mascheroni during his study of Euler’s Gamma function. The difference caught the eye of Gauss given his interest in $\textrm{Li}(x)$ and his own computation recovered von Soldner’s value till the 22nd place. Gauss then called upon his 19 year old student, the astronomer Friedrich Nicolai, who was a mental calculating prodigy, to calculate Euler’s constant using a method based on Euler’s own sums. He did so to 40 decimal places showing von Soldner to be correct. As a result of von Soldner’s tables we got the first glimpse of how $\textrm{Li}(x)$ looks (Figure 2). Von Soldner also realized at one value of $x$ the value of $\textrm{Li}(x)=0$ while being apparently unaware of its importance in the matter of the prime number distribution.

Figure 2 with the Ramanujan-Soldner point

Unaware of most of the work done in the West over the past century, Srinivasa Ramanujan carried out his own studies to independently obtain expressions for the prime number distribution. In the process he discovered for himself that $\textrm{Li}(x)$ approximated the prime counting function. Ramanujan discovered multiple fast-converging series to evaluate $\textrm{Li}(x)$. The simplest of which is:

$\textrm{Li}(x) = \gamma + \log(\log(x))+\displaystyle \sum_{k=1}^\infty \dfrac{\log(x)^k}{k!\cdot k}$, where $\gamma$ is Euler’s constant.

In studying $\textrm{Li}(x)$ Ramanujan also independently discovered the approximate value of $x$ for which $\textrm{Li}(x)=0$ (Figure 2) and mentioned it in his exchange with Hardy. Thus, it is now known as the Ramanujan-Soldner constant: 1.45136….

To gain further appreciation of Ramanujan’s independent discovery of the use of $\textrm{Li}(x)$ in approximating $\pi(x)$ we shall step back to the studies of Riemann in this regard. Riemann wondered if he could get a better approximation of the prime counting function $\pi(x)$ than $\textrm{Li}(x)$ and arrived at the following expression:

$\pi(x) = \textrm{Li}(x) - \dfrac{\textrm{Li}(\sqrt{x})}{2} +t_1+t_2...$

Since we are keeping this note very basic we shall not expand on the additional terms $t_1, t_2...$, but the $t_1$ term is related to the zeros of the Riemann’s $\zeta$ function in the complex plane. However, by using just the first listed term alone we get the fit shown in Figure 3.

Figure 3 $\pi(x)$ in red, $\tfrac{x}{\log(x)}$ in blue, $\textrm{Li}(x)$ in green and $\textrm{Li}(x) - \dfrac{\textrm{Li}(\sqrt{x})}{2}$ is in black.

One notices right away that it is a better approximation in the range shown and on average is better than just $\textrm{Li}(x)$. Ramanujan independently arrived at the same expression, short of the additional term dependent on the complex plane zeros of the Riemann $\zeta$ function, and mentioned it to Hardy in course of their early exchange of letters. In his expression he captured that fact that $\textrm{Li}(x)$ does not count the prime numbers per say but the integer powers of primes weighted by $\tfrac{1}{n}$. Thus, the first term $\tfrac{\textrm{Li}(\sqrt{x})}{2}$ corrects for the squares of primes; similarly in Ramanujan’s equivalent expression the term $\tfrac{\textrm{Li}(\sqrt[3]{x})}{3}$ would correct for cubes of primes and so on.

From figure 1 and figure 2 we see that $\textrm{Li}(x)> \pi(x)$ in the range we have plotted it. Indeed, the fact that the negative term $-\tfrac{\textrm{Li}(\sqrt{x})}{2}$ results in a better fit to $\pi(x)$ is in line with this. This led Gauss and Riemann to wonder if indeed $\textrm{Li}(x)$ was always greater than $\pi(x)$. It appears Ramanujan also initially might have thought the same and saw the negative terms as “bringing down” $\textrm{Li}(x)$ to $\pi(x)$. This indeed remains true for all $n$ that has been within our computational reach to date. However, remarkably, Littlewood produced a proof that $\textrm{Li}(x)$ and $\pi(x)$ cross each other infinite number of times. This proof is rather difficult for those of meagre mathematical capacity, but it indicated that there should be some very large number where $\pi(x)$ will, for the first time, become greater than $\textrm{Li}(x)$. This can be intuitively understood in terms of the above expression for $\pi(x)$ in terms of $\textrm{Li}(x)$ provided by Riemann. As noted above, beyond the terms that correct for the powers of primes (the biggest being $-\tfrac{\textrm{Li}(\sqrt{x})}{2}$), there is the term based on the complex-plane zeros of the Riemann $\zeta$ function. While normally that term reduces to something negligible, at some big number that term could neutralize the $-\tfrac{\textrm{Li}(\sqrt{x})}{2}$ term and make $\pi(x)$ bigger than $\textrm{Li}(x)$. We have not reached that number through direct computation but Littlewood’s student Skewes came up with the first estimates of how big that number could be. Since then there has been a very active program in computational mathematics to size up that number. When I last checked it was established that the first crossing of $\textrm{Li}(x)$ and $\pi(x)$ would happen somewhere near the gargantuan $1.397162914 \times 10^{316}$. This is truly an example of how computational intuition gained from even large numbers can eventually fail at some humongous number out of the reach our computation.

In addition to the above Riemann also discovered yet another prime counting function that he thought would be more or less the same as $\pi(x)$. This function $\textrm{Ri}(x)$ combines the Möbius function $\mu(x)$, which we described in the previous note and $\textrm{Li}(x)$:

$\textrm{Ri}(x)= \displaystyle \sum_{n=1}^\infty \dfrac{\mu(n)}{n}\cdot\textrm{Li}(x^{1/n})$

Figure 4. Part 1: $\pi(x)$ is in red, $\textrm{Ri}(x)$ in black and $\textrm{Li}(x)$ in green; Part 2: $\pi(x)-\textrm{Ri}(x)$

In Figure 4 we calculated $\textrm{Ri}(x)$ as an approximation using the sum of the first 1000 terms in the above expression. This gives a value correct to the first place after the decimal point. This enough for our current purpose but is computationally costly as you need 1000 terms just to get this level of accuracy. As can be seen in Figure 4, $\textrm{Ri}(x)$ closely approximates $\pi(x)$ in the range we have plotted. Strikingly, Ramanujan entirely independently of Riemann arrived at the same function as $\textrm{Ri}(x)$ in course of his studies on the prime number distribution before he began his exchange with Hardy.

As we saw, the original definition of $\textrm{Ri}(x)$ takes a lot of terms to get even limited accuracy. In the late 1800s the Scandinavian mathematician J.P. Gram discovered a series $\textrm{G}(x)$ that is equivalent to $\textrm{Ri}$ using the Riemann $\zeta$ function:

$\textrm{G}(x)=1+\displaystyle \sum_{k=1}^\infty \dfrac{(\log(x))^k}{k \cdot k! \zeta(k+1)}$

Using the above series with just 99 terms we can compute $\textrm{Ri}(x)$ to at least 5 places beyond the decimal point. Interestingly, Ramanujan arrived at yet another fast-converging series (we will call it $\textrm{Rjn}(x)$ after him) on his own to approximate $\pi(x)$, which gives values close to $\textrm{G}(x)=\textrm{Ri}(x)$ (Figure 5, top panel). Being the master of the Bernoulli numbers, which have an intimate connection with the Riemann $\zeta$ function, Ramanujan used them instead of that function.

$\textrm{Rjn}(x)= \dfrac{4}{\pi} \displaystyle \sum_{k=1}^\infty \dfrac{(-1)^{k-1}\cdot k}{(2k-1)\cdot B_{2k}} \cdot \left(\dfrac{\log(x)}{2\pi}\right)^{2k-1}$ where $B_{2k}$ are even Bernoulli numbers.

Ramanujan did not stop there and went on to produce a further approximation of $\pi(x)$ as a definite integral, which we will denote as the Ramanujan integral $\textrm{Rji}(x)$:

$\textrm{Rji}(x)=\displaystyle \int_0^\infty \dfrac{(\log(x))^t dt}{t \cdot \Gamma(t+1) \cdot \zeta(t+1)}$ where $\Gamma(x)$ is the Gamma function.

This integral too gives values close to the series $\textrm{Rjn(x)}$ and $\textrm{G}(x)=\textrm{Ri}(x)$ as shown in Figure 5.

Figure 5. In the top panel the $\textrm{Rjn}(x)$ was calculated using the first 15 terms, which are sufficient to give convergence to 5 places after the decimal point. In the bottom panel $\textrm{Rji}(x)$ was approximately calculated using the Gauss-Kronrod numerical integration method with the upper limit of the integral taken as 150 due to limitations in evaluation beyond that.

As we saw above (Figure 4) $\textrm{Ri}(x)=\textrm{G}(x)$ approximates $\pi(x)$ better than $\textrm{Li}(x)$ in the range we have plotted it (the plot will be practically identical for the Ramanujan series and integral given that their values are close to $\textrm{Ri}(x)$: Figure 5). This indeed has been shown to be true computationally for $n$ even 10000 times bigger than our plotted range. Thus, it is not surprising that both Riemann and Ramanujan thought that $\textrm{Ri}(x)$, $\textrm{Rjn}(x)$, $\textrm{Rji}(x)$ are much better approximations of $\pi(x)$ than $\textrm{Li}(x)$ or more or less $\pi(x)$ itself. However, here again numerical intuition was shown to be imprecise. In a publication by Hardy and Littlewood, the latter remarkably proved that starting at some large number $\textrm{Li}(x)$ would become a better approximation of $\pi(x)$ than these functions of Riemann, Gram and Ramanujan. Further, they would swap positions as the better approximation of $\pi(x)$ an infinite number of times in the realm of giant numbers. Thus, one can only say that:

$\textrm{Ri}(x)=\textrm{G}(x) \sim \pi(x);\; \textrm{Rjn}(x) \sim \pi(x); \; \textrm{Rji}(x) \sim \pi(x)$

Ramanujan did all this work in isolation at Kumbhaghoṇa and Chennai before leaving for Cambridge. It must be remembered that he did have access to any serious cutting edge literature and his primary inspiration was evidently provided by Carr’s synopsis. Thus, after reading Carr, at age 17 Ramanujan independently discovered the famous Bernoulli numbers, which were originally discovered by Jakob Bernoulli (and perhaps by the Japanese Samurai Seki Takakazu or someone of his school) and were subject of much investigation in the west. Armed with this he went on to calculate Euler’s constant $\gamma$ to 15 places after the decimal point, thus coursing on the path taken by Euler. This formed the platform for his foray into the distribution of prime numbers. While this has been routinely described as nonrigorous work starting from Hardy’s initial letters to Ramanujan, it does bring out his extraordinary intuition. To us this work established his connection to the past greats, even as Littlewood declared to Hardy that he was in the league of Euler or Jacobi (Or Bertrand Russell: “I found Hardy and Littlewood in a state of wild excitement because they believe they have found a second Newton, a Hindu clerk in Madras making 20 pounds a year.“). Indeed, it is rather striking how he single-handed covered some of the key ground spanned by the discoveries of Jakob Bernoulli, Euler, Gauss, Dirichlet and Riemann. He rediscovered many of their intuitions regarding the prime counting function. Not just that, he came up with his own with expressions, sometimes better the previously published ones, for the Bernoulli numbers, $\textrm{Li}(x)$, $\textrm{Ri}(x)$ and other asymptotic expressions for $\pi(x)$. Short of the Riemann $\zeta$ function in the complex plane with its non-trivial zeros, Ramanujan penetrated key elements of the great prime number question all by himself. Thus, he was like one man all by himself trying to cover the “gap” in the mathematics of the Hindus following its decline from the high-point reached by the nambūtiri-s and their school. In many ways, while operating in modern mathematics, he seemed mysteriously “channel” the characteristics of the old Hindu greats, with their love for approximating functions and fast-converging series, which they delighted in since the age of the Yajurveda.

If Ramanujan represents the mastery of intuition, the other side, that certainty comes from rigor, is shown by the striking proofs of Littlewood regarding the $\textrm{Li}(x)$ and other approximations and $\pi(x)$. This was exemplified by Gauss, who, though unrivalled at intuition and experimentation, strove for rigor. He did not publish many of his works that contained key intuitions if he was unable to produce a proof for them as exemplified in his dense and rigor-filled Disquisitiones Arithmeticae. This aspect of mathematics, which is especially seen in higher arithmetic, is what sets mathematics apart from science. In science, barring those aspects which are truly reduced to a mathematical abstraction, we have nothing like a proof. Instead, a hypothesis is repeatedly tested for falsification. Thus, in scientific setting, things like $\textrm{Li}(x)> \pi(x)$, or as we saw in the previous article the Mertens function being inside the $\sqrt{n}$ parabola, would have survived all attempts at falsification by direct experimentation (in this case computational). Yet, in mathematics they can be ultimately falsified even though they lie beyond the reach of direct experimentation.

Nevertheless, the fact that Ramanujan, far removed in space and time, was able to recapitulate and out do results of other great mathematicians before him suggests that mathematics is by no means an invention of the human mind. Rather, Ramanujan, like Euler, or Gauss or Riemann before him, were tapping into that Platonic realm where the mathematical ideals exist. The human intuition might grasp only “smudged” image of the ideal, yet because it exists even those images would be similar when apprehended independently by different explorers. Then the apparatus of rigor might get one closer to the ideal, but only after the intuition has apprehended it in the first place.

Posted in History, Scientific ramblings |

## Our auto-discovery of the Möbius and Mertens sequences

Recently, we were explaining to our friend the Möbius and the Mertens functions and their relationship to the prime number distribution. We also heard with some wonder from a physicist of a theoretical model where multiparticle states behave as bosons or fermions in way predicated by the Möbius function. This prompted us to put down the tale of our autodiscovery of the Möbius and Mertens sequences in our youth. On one hand that journey gave us some satisfaction that we, in a very limited way, were on our own able to recapitulate the journey of the great minds. But, at the same time, it also brought us down to earth showing how little we were able to do on our own following the rediscovery of their results, thus, telling us how the truly great and pedestrian are distinct.

As we have described before, we began our exploration of sequences by hand and computer inspired by Hofstadter’s book (GEB). One of the sequences that we conceived in course of this exploration was a one-seeded sequence which goes thus in words: We start with 1 as the first term. Then for every integer $k$ from 1 to $n-1$ we test if it divides $n=2,3,4...$. For each $n$, we then take the negative of the sum of all the $f[k]$, where $k$ satisfies this divisibility condition. This becomes the value of the nth term of the sequence $f[n]$

$f[1]=1$
$\displaystyle f[n]= -\sum_{k} f[k]$ where $n \mod k \equiv 0, 1 \le k < n$

We were at first surprised to see that this sequence takes only 3 values -1, 0, 1. The first 25 terms are: 1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, -1, 1, 1, 0, -1, 0, -1, 0, 1, 1, -1, 0, 0

Figure 1. A plot of the first 2000 terms (the red short lines are $f[n]=0$).

The sequence (Figure 1) at first looked almost random but after looking more closely we began to discern a pattern. First we asked where do the 0s come? After analyzing their locations it became clear that they came wherever $n$ is a perfect square of a prime number or a multiple of such a perfect square. Thus, $f[n]$ where $n$ is 4, 8, 9, 12, 16, 18, 20, 24, 25… are all 0s. We then asked where the -1s and 1s come. It was immediately clear that when $n$ is a prime $f[n]=-1$. Why this should be so is obvious from the way we defined our sequence. We also noticed that whenever a number has a odd number of prime factors then $f[n]=-1$. Thus $n=30=2 \times 3 \times 5; \; f[30]=-1$ or $n=42=2 \times 3 \times 7; \; f[42]=-1$. For all the remaining values of $n$ we get $f[n]=1$. These values of $n$ have a even number of prime factors. Thus, for $n=6=2 \times 3, 21=3 \times 7$ we get $f[n]=1$. One learns from these patterns how the $f[n]$ values for a given $n$ can be understood on the basis of the sequence definition.

These patterns also directed us to render them graphically as a square matrix with its total number of cells amounting to a certain perfect square $n$ where each cell takes one of three colors based on the value of $f[n]$. By playing with different dimensions we arrived at $n=125 \times 125$ for capturing the above-described structure of this sequence (Figure 2). The 5 vertical lines are generated by the square number 25 and its multiples. The square number 4 and its multiples create the $135^o$ one-square cross-lines. 9 and its multiples create the $45^o$ cross-lines. When multiples of 9 and 4 come adjacent to each other e.g. (8,9), (27,28), (44,45), (63,64), (80,81), they create the broad 4-square cross-lines. They have an alternating prime number separation of 19 and 17.

Figure 2. $f[n]$ for $n=125 \times 125$ with the first row from bottom being $n=1:125$. Brown: -1; light yellow: 0; green: 1

This pattern led us to ask the question as to what fraction of the area of a square, like the above one, is one of the three colors. Experimentally, it is obvious that the 0s occupy most area, while the remainder is evenly split between -1 and 1. Figure 2 helped us to arrive at approximate value for the area occupied by the zeros. For the first row $n=1:125$, counting repeat numbers only once, we get: 31 numbers containing 4 as a factor; 10 with 9; 4 with 25; 2 with 49 and 1 with 121. This adds to 48. Thus for the first row of figure 2 the fraction of 0s is $\tfrac{48}{125}$. Given that these smaller squares set the basic patterns in figure 2, and as we climb up the matrix we add additional perfect squares (e.g. $13^2=169$ between $n=126:250$) and their multiples, we added 1 more to the numerator to arrive at the fraction of 0s $\approx \tfrac{49}{125}= 0.392$. The experimentally determined value oscillates in the vicinity of that value with a tendency for convergence with increasing $n$ (Figure 3).

Figure 3. The gray line in the top panel is $1-\tfrac{6}{\pi^2}$ while in the bottom panel it is $\tfrac{3}{\pi^2}$

This value to which the fraction of 0s converges immediately caught our eye because of a book we had just borrowed from the local library on elementary number theory. It informed us about the masterful discovery of the great Leonhard Euler: What is the probability that any two positive integers chosen at random are mutually prime (i.e. have GCD=1 or are coprime)? The solution to this problem is related in turn to what was called the Basel problem after the hometown of Euler and the Bernoullis. The Basel problem asks the sum of the infinite series:

$1+\dfrac{1}{4}+\dfrac{1}{9}+\dfrac{1}{16}+\dfrac{1}{25}...= \displaystyle \sum_{n=1}^\infty \dfrac{1}{n^2}$

The 28 year old Euler showed that this sum is $\tfrac{\pi^2}{6}$, thus, surprisingly, bringing in the number $\pi$. Now the answer to the original question is the inverse of this number. Thus, the probability that two randomly chosen integers are coprime is $\tfrac{6}{\pi^2} \approx \tfrac{76}{125}$. This, result, even more miraculously, connects $\pi$ to the relatively prime numbers (Figure 4).

Figure 4. The matrix of relative primality of pairs of integers for $n=1:100$. The area in red to the total area of the square converges to $\tfrac{6}{\pi^2}$

That approximate value we list above immediately led us to realize that our fraction of the area under 0s in figure 2 or, more precisely, the asymptotic fraction of 0s in our above sequence $f$ should be exactly $1-\tfrac{6}{\pi^2}$. This also gives the fraction of numbers containing perfect squares in $1:n$. Conversely, the fraction of -1s and 1s in our sequence are $\tfrac{3}{\pi^2}$. This gives the fraction of square-free numbers, which are products or even or odd number of primes in $1:n$ Thus, an exploration of a sequence simply out of the curiosity of checking out patterns emerging from divisibility led us to a deep link to primality and $\pi$.

Given that the fraction of -1s and 1s are approximately the same we wondered what might be the trends in the sum of the above series $f$. Accordingly we defined new series thus:

$f1[n]=\displaystyle \sum_{k=1}^n f[k]$

Figure 5 shows a plot of $f1[n]$ for $n=1:30000$

Figure 5

It is clear that $f1[n]$ oscillates between negative and positive values in a what looks like random walk. However, we do notice that as $n$ increases the extreme values reached by $f1[n]$ are greater (Figure 6).

Figure 6

We wondered what this rate of increase might approximated by. But that was where we hit the end of our explorations. Those days the internet was not yet available to us and Riemann hypothesis was not the kind of thing that was widely known in lay circles. We knew of it and the famed Riemann $\zeta$ function but nothing much of the intricacies of this hypothesis. It was then that in a city teeming with Meccan demons we briefly borrowed a dense tome on the number theory from a distant relative of ours who was belonged to a clan with multi-generational mathematical talent. It was there that we learned that our auto-discovered sequence $f$ was something that was first described by the prince of the mathematics, Carl Gauss, in his late teens or early twenties and published in his famed Disquisitiones Arithmeticae. However, it came be known after a student of his, August Möbius, who rediscovered and studied it more than 30 years later. This Möbius function can be defined thus:

$\mu(n)=\begin{cases} 0; \textrm{n contains square of a prime}\\ 1; n=1\\ (-1)^k; \textrm{n is product of k distinct primes} \end{cases}$

This $\mu(n)$ has a relationship to the Riemann $\zeta$:

$\displaystyle \sum_{n=1}^\infty \dfrac{\mu(n)}{n^s}= \dfrac{1}{\zeta(s)}$ where $s \in \mathbb{C}$

The fraction of 0s in our sequence $f$ can be expressed as converging to $1-\tfrac{1}{\zeta(2)}$ and the fraction of 1s and -1s as converging to $\tfrac{1}{2\zeta(2)}$

Another interesting property of $\mu(n)$ is seen when it is included in a series of Lambert; we get this interesting sum:

$\displaystyle \sum_{n=1}^\infty \mu(n) \cdot \dfrac{x^n}{1-x^n}=x$ where $|x| <1$

The summatory sequence which we described above as $f1$ turned out to be a function discovered by Mertens $\textrm{M}(x)$ in the late 1800s. Mertens, after computing the first $10^5$ values by hand, boldly conjectured that $\textrm{M}(x)$ will for all $n$ lie inside the parabola $y=\sqrt{x}$, shown in blue in Figure 4. But this conjecture was shown to be false in our times by Odlyzko and te Riele. However, the first value for which $\textrm{M}(x)$ will out grow the parabola is so large that it is way outside our current computational reach. But weaker expressions related to this conjecture are shown to be equivalent to the Riemann hypothesis, which we will assume the reader is familiar with:

$M(x)=O(x^{1/2+\epsilon})$ where $\epsilon >0$

To briefly explain the big $O$ notation for an unfamiliar reader: Let there be two functions $f(x)$ and $g(x)$ whose behavior we are comparing as $x \to \infty$. In our current case $f(x)$ is $M(x)$ and $g(x)=x^{1/2+\epsilon}$.
If $|f(x)| \le K\cdot |g(x)|$ for all $x \ge x_0$, where $K$ is a positive real constant and $x_0$ is some real number, then we may state that $f(x)=O(g(x))$.

The link to the prime distribution also comes out in a nice relationship discovered by Mertens between $\textrm{M}(x)$ with the functions defined by the famed Russian scientist Chebyshev in his study of this problem (figure 7). The first of Chebyshev function $\Theta(n)$ is defined thus:

$\Theta(n)= \displaystyle \sum_{p \le n} \log(p)$ where $p \le n$ are all primes less than or equal to $n$

The second Chebyshev function $\psi$ is defined thus:

$\psi(n)=\displaystyle \sum_{p \le n} \left \lfloor \log_p(n) \right \rfloor \log(p)$ where $p \le n$ are all primes less than or equal to $n$

Figure 7. The top panels show the growth of the two Chebyshev functions. The bottom left panel shows the convergence $\tfrac{n}{\Theta(n)} \to 1$. The right bottom panel shows the fluctuations of $\psi(n)-n$ reminiscent of $\textrm{M}(x)$.

This $\psi(n)$ is related to the $\textrm{M}(x)$ thus:

$\psi(n)=\displaystyle \sum_{k=1}^n \textrm{M}\left (\left \lfloor \dfrac{n}{k} \right \rfloor \right ) \cdot \log(k)$

We first computationally demonstrated this relationship for ourselves when we learned that our $f1$ was none other than $\textrm{M}(x)$. Thus, with just a household computer, working knowledge of a modern computer language, and some school and elementary college level numeracy one can at least get a glimpse of some of the deepest issues in mathematics. Visualizing some of these, even at a very low level as we have done, gives you a mystical experience — i.e. a glimpse of some of deep mysteries of existence.

Posted in Life, Scientific ramblings |

## Mongolica: Knowledge preservation and generation, Bolad Aqa and the like

We had earlier written an essay on the preservation and production of synthetic knowledge by the Chingizid Mongols. Here we discuss a few additional points in that regard.

It is clear that throughout the Chingizid clan there was a certain proto-scientific interest right from the beginning. Astronomy was one area that appears to have interested many of them. It perhaps relates to the fact that the full expanse of the sky leaves a profound impression on the observer on the vast openness of the steppe. Indeed, in Mongol shamanic oral tradition there are accounts such as Ursa Major being the banner of the tngri-s. Thus, starting from Chingiz Khan we see considerable interest in patronizing astronomers. In 1218 CE the Chingiz Khan recruited the gigantic, long-bearded Yelü Chucai (said to have been ~2.05 m in height), who came from the earlier Mongolic branch, the Khitan. While Yelü Chucai was an all-round scholar, the great Khan was particularly interested in his knowledge of meteorology and astronomy. The Khan used his knowledge to predict snow late in year on the steppes during his Khwārizm campaign. He also predicted eclipses for the Mongols and showed the Khan a comet towards the end of the campaign in Turkestan, which is likely to have been the 1222 CE apparition of Halley’s comet. Not surprisingly these scientific observations of Yelü Chucai were mixed with a form of Sino-Mongolic astrology. After the conquest of Samarqand, Chingiz Khan set up an observatory there and appointed a cīna astronomer Li, likely at Yelü Chucai’s behest. In course of the Turkestan campaign Yelü Chucai obtained astronomical works of the Hui-ho [from the Mohammedan world] and studied them closely. As a result he detected errors in the Chinese method of calculations and created a revised method for the Mongol calender and astronomical tables.

The governor of Tibet in the mid-1300s, Barandnā (Skt: Prajña), the descendant of Qubilai Khan, also commissioned extensive astronomical works deriving from Hindu and Hellenistic traditions that led to Mongolian commentaries being prepared on them. This Hindu astronomy was transmitted in part via the last great tantra of the nāstika-s, the Kālacakra, which had elements modeled after the now lost eponymous saura tantra. Some of this material survives in the ritual manual preserved by Sangwar, a Mongol official in the Manchu times from Boro Balgasun in Ordos. Here we see a legendary Khan receiving Indian calenderical knowledge from goddess Vimā, the daughter of the great Indra. There is a table preserving the old Indian nakṣatra system: “naghšidar-un ner-e odun-u togh-a bui düri ba maqabud ene bu”: The nakṣatra names, number of stars, figure and element. We also see the incidental transmission of ritual of ultimately Hindu origin in the process such as: “basa nigen eketü Mahašuvari tngri”: Another topic concerning the god Maheśvara; “Maqagala baghuqu edür anu”: As for the day Mahākāla descends; “Okin tngri baghuqu edür kemebesü”: As for the day the goddess Śri descends.

Outside of the “Golden family” there were some Mongol intellectuals who played an important role in this knowledge venture of the Khans. One extraordinary individual who was involved in this process was Bolad Aqa. We wished to detail his history in our earlier note centered on Rāshīd ad-dīn but wanted to read more sources pertaining to him; so we left out that part of the story then. More over, a good review on his history role was also provided by TT Allsen. In this part of the note we shall offer a brief summary of his history. In the Secret History the origin myth of the Chingizid Mongols states that they were descendants of the wolf Börte Chino and the doe Qo’a-maral at the holy slopes of the Burqan-qaldun by the Onon river. From them eleventh in descent were the two brothers Du’a Soqor and Dobun Mergen. Du’a Soqor was said to have a single cyclopean eye by which he could see a great distance. He found a vast slab of nephrite which is said to have been his throne [Later Ulugh Beg is said to have obtained the same from Mongolia and embellished Timur’s grave with it]. From the top of Burqan-qaldun he is said to have seen a beautiful woman Alan Qo’a in a camp at a great distance and asked his brother Dobun Mergen to seek her as his wife. Alan Qo’a became the legendary ancestress of the Chingizid Mongols. But her sons are said to have had a rift with the four sons of Du’a Soqor and they parted ways. While Chingiz descended from the former, the latter are said to have formed the four clans, the Dörben irgen. They were ranged against Temujin during his rise. Aligning with the Tayichi’ud they fought him but were forced flee when he destroyed that alliance. Then the Dörben joined Jamuqa against Temujin but he decisively demolished that alliance in 1202 CE. Then they joined the tayang Khan of the Naiman Turks against Temujin but even that alliance was smashed by him in 1204 CE. At this point they surrendered to him and joined his ranks.

One of them, Yürki won Chingiz Khan’s trust and he was appointed as the ba’urchi (chef) of the Khan’s main wife Börte. In addition he served a military role as commander of a hundred men in a force of 1000 that was directly commanded by the Khan himself. His son was Bolad Aqa, who inherited his position as a royal chef under Qubilai. The prestigious position as the ba’urchi allowed him to receive education along side Qubilai’s sons. In course of this, his intelligence became apparent and he was found to have great skill at translating extemporaneously between Chinese and Mongolian. Thus, he was given the additional responsibility of dealing with the Chinese officials who were being absorbed into the Mongol administration. He also likely knew some Sanskrit for he is said to have liaised with Kashmirian Bauddha Tāntrika-s and trained some of them for joining the Qubilai’s personal corps. Bolad was also given the task of adapting cīna rituals as part of the ceremonies for the coronation of Qubilai as the Kha’Khan. As the ba’urchi he also performed the four animal-sacrifices (like the Indo-Aryan śamitṛ in the Vedic animal sacrifice), which the Mongol Khans conducted to prepare the meat offerings to the tngri-s and ancestors. He also supervised the preparation of beverages for the royal family.

In the east itself, at the age of ~33 years, Bolad was called upon by the Khan to set up the imperial library. He along with the Chinese assistant Liu Ping-chung put together a body of archivists, historians and other collectors of information. TT Allsen suggests that this library collected a body of books, maps and pictures. Such a collection of information appears to have been the basis of the parallel project conducted by Bolad in the west along with Rāshīd ad-dīn and others. This project might be summarized in the words of Rāshīd ad-dīn thus:

Today, thanks to God and in consequence of him, the extremities of the inhabited earth are under the dominion of the house of Chingiz Khan and philosophers, astronomers, scholars and historians from North and South China, India, Kashmir, Tibet, [the lands] of the Uighurs, other Turkic tribes, the Arabs and Franks, [all] belonging to [diﬀerent] religions and sects, are united in large numbers in the service of majestic heaven [translation in to Persian of Mongolian Möngke Tngri]. And each one has manuscripts on the chronology, history and articles of faith of his own people and [each] has knowledge of some aspect of this. Wisdom, [which] decorates the world, demands that there should be prepared from the details of these chronicles and narratives an abridgement, but essentially complete [work] which will bear our august name.”

[cited from translation of Rāshīd ad-dīn provided by TT Allsen based on the Moscow edition]

This effort of bringing together knowledge form different parts of Asia on part of the Mongols had different consequences on different people. In the west, it led to the reintroduction of Hellenistic knowledge via the Mohammedan sources to the Europeans, thereby greatly contributing to their scientific revolution. In the Mohammedan world itself this knowledge showed patches of revival and survival when not smothered by the hand of Mohammedanism – e.g. the case of Ulugh Beg. The cīna-s, while participating in this process when under Mongol rule never seemed to have realized the value of Hellenistic knowledge and internalized it until closer to our times. Among the Mongols themselves, upon the fall of their empire such interest appears to have mostly waned with apparently only limited revival in later times based on Hindu astronomy filtering via Tibet, among other things via revival of the Kālacakra tradition.

For comparison, we shall take a detour to consider the case of the old Indo-Greek interactions. This interaction was not carried out under any active royal supervision. We have some evidence that Alexander despite his antipathy towards brāhmaṇa-s, who had galvanized the kṣatriya-s to wipe out the yavana invaders in Bhārata, had some personal interest in Indian philosophers and scientists and conveyed some of the know-how of the later to Greece (e.g. how Ptolemaios was sedated during a surgery by Sarpagandha). Subsequently, there was clearly an interaction between the two traditions. On the Hindu side, the Hindu astronomers honestly state that they closely studied Greek astronomical works that they could lay their hands on and produced multiple commentaries and Sanskrit reworkings of them. On the Greek side it is less clear from their own sources as to what, if anything they received from the Hindus. Hence, a wide range of opinions have been proffered by modern Euro-Americans in particular. At one end of the spectrum we have Peter Green who thinks that while the Hindus and yavana-s met in years following the Macedonian invasion nothing much really came of it in terms of knowledge transmission. At best both sides poorly understood or forgot the other. On the other end we have the archetypal white Indologist Pingree and followers (“Hindus as idiots”) who believe that everything meaningful in Hindu science came from the yavana-s, so much so that there was no real Hindu mathematics of note before being enlightened by the yavana-s. Others like McEvilley have accepted that Hindu philosophical and perhaps medical knowledge might have been transmitted to the yavana but mostly for mathematics it was the other way around.

After considering the evidence ourselves we feel that all kinds of scientific knowledge was transmitted both ways but it was not entirely incorporated into respective systems. For instance, in the case of astronomy and mathematics the Hindus while studying and reworking the yavana works into Sanskrit never incorporated the material into their core models. Āryabhaṭa’s heliocentric background for the model is unlike that of any Greek work despite the fact he was aware of them and he and his students studied them. Hence, some Euro-American workers try to claim that it should have come from a lost Greek model without presenting evidence for any such in the Greek world. That simply suggests that it represents a purely Hindu development. The real core of Greek mathematics in the form of Euclidean geometry was never adopted by Hindus, but the pseudo-scientific material of yavana astrology made a deep impact on Hindu astrology. On the Greek side they of course adopted several aspects of Hindu medicine and philosophy but what about mathematics. We hold (as some early modern Europeans did in more innocent days) that they acquired a bit of Hindu algebra but on their part never really understood its spirit and remained limited in its development in Heron’s and Diophantos’ works. Thus, the two had rather distinct mathematical frameworks that were apparently not able incorporate material from the other at any deep level perhaps in part because the deepest of this work did not travel widely between the civilizations unlike in the case of astronomy. In the case of the Mongol empire while the patrons brought together very distinct knowledge systems it appears that the synthesis did not filter down in a big way in China and Korea, though it might have made more of an impact, even if indirect in the west.

Posted in History |

## Civilizational collapse, complexity, innovation and neomania

On these pages we had provided some commentary on the work of the German thinker Oswald Spengler and his ideas of civilizational development. We had also provided other perspectives on this matter derived from analogies stemming from simple mathematical systems. We originally wished to detail other thoughts in this direction, particularly relating to civilizational collapse, within the conversational framework we have often used on this pages. However, not everything the frail mortal proposes is achieved, as the mighty gods, who show reveal glimpses of deeper nature of existence to him, also shower afflictions that dispose of his plans. Hence, we simply decided to place a few words relating to this topic here rather than work it into a piece in the series. In part this was triggered by the “neomania” I heard from people reaching a crescendo, all enthusiastic about the solution to deep human problems by new technologies that were just around the corner. Here, I am using neomania in a sense I have seen the Lebanese author NN Taleb use it – an unbridled sense of desire and hope for new technology as a panacea. It was ironic that two of the biological technologies that were talked about involved me as a discoverer – all I got from them was a sense of pleasure of uncovering a hidden insight at the time I stumbled upon those discoveries – beyond that they have not solved any of my own human problems. Hence, perhaps one could call me biased in this regard. However, I should clarify that even I would welcome some of the neo-technologies and hope they improve my life but I remain a skeptic regarding them ultimately solving any fundamental problems or being better *on an average* from less-glamorous solutions. All of this led to some thinking about the book by the American historian Tainter, which is one of the more detailed accounts of the endgame or civilizational collapse (Collapse of Complex Societies).

He cites Casson regarding the aftermath of the termination of Pax Romana in Britain: “From A.D. 100 to 400 all Britain except in the north was as pleasant and peaceful a countryside as it is to-day … But by 500 A.D. it had all vanished and the country had reverted to a condition which it had, perhaps, never seen before. There was no longer a trace of public safety, no houses of size, dwindling townships and all the villas and most of the Roman cities burnt, abandoned, looted and left the habitation of ghosts

This kind of end is not unique. Tainter collects several examples: 1) the western Chou empire in China; 2) Harappan civilization in India; 3) The Mesopotamian kingdoms; 4) Egyptian Old and Middle kingdoms; 5) The Hittite empire; 6) Minoan Civilization; 7) Mycenaean Greek civilization; 8) The Roman Empire; 9) The Olmec in Mexico; 10) The Teotihuacan civilization in Mesoamerica; 11) Casas Grandes near the Mexico/USA border, the Chacoans of New Mexico and the Hohokam of Arizona (these three adjacent native American cultures probably had some interactions but it is not clear as what the exact relationship between them was); 12) the native American culture of the Cahokia mound complex in the middle Mississippian zone; 13) Huari and Tiahuanaco kingdoms of Andean South America.

One can debate some of the cases of collapses collected by Tainter. One may also say his list is heavy on the old Americans – not surprising given that he is an American historian. For example, right in Asia there are other notable examples, like the Khmer civilization, which showed the classic pattern of efflorescence followed by collapse. It is also pretty weak on sub-Egyptian Africa. Immediately south of Egypt, the Meroëtic civilization can be mentioned as an independent case. The sub-Saharan Great Zimbabwe can also be considered another valid African example. We could also consider the Benin empire, although one could place the primary blame for it is collapse on European Christian contact. Nevertheless, we do feel Tainter’s is generally a good list. Further, the Americo-centricity is not entirely a bad thing thing because the Americas developed for long without significant contacts with Eurasia after the initial peoplings, Thus, they were a good independent laboratory to study the development of human organizational complexity. In these examples Tainter makes the case for development from a relatively simple organizational state to a complex state that was characterized by increasing organizational complexity followed by a “golden age” and then a collapse, where most features of organizational and technological complexity are lost and the population returns to a basic subsistence mode.

One may ask how does one measure complexity of a civilization? We would say a relatively simple device, inspired by devices we have effectively used in our work in biology, can capture social complexity: Let us denote each distinct occupation performed as primary source of livelihood by an individual in a society as a character. Then per household we can create a string of such characters concatenating the character representing each individual in the household. Then by aggregating those across all households in the human habitation or civilization under consideration we get an occupation string for the entire habitation/civilization. We then calculate the Shannon entropy for this string. This will be a reasonable measure of complexity. Such a measure does suggest that as a civilization or culture develops it rises in complexity reaching a peak value. While this peak value might have considerably differed between the Eurasian+Egyptian-zone civilizations and those of deep Africa and the Americas the trend in its value is the same in course the development of a culture or civilization followed by a collapse.

As another dimension we may also consider the recent work by TA Kohler and colleagues (Greater post-Neolithic wealth disparities in Eurasia than in North America and Mesoamerica by Kohler TA et al). They used archaeologically determinable house-size distribution to compute Gini coefficients. The Gini coefficient is a measure of the inequality in the values comprising a frequency distribution (g=0, total equality; g=1, one owns all and rest nothing) and is typically used to measure the inequality in income distribution in a society or habitation. They showed that the Gini coefficients increased with the domestication of plants and animals in the Neolithic and subsequently with the emergence of the Indo-European model of wide-ranging conquests over Eurasia. In their limited analysis they found that the disparity indicated by the Gini coefficient was more in parts of Europe and Asia they sampled than in the parts of the Americas they studied. The average Gini coefficient was markedly higher in cultures with classic states than those that remained restricted to the local habitations.

An implication of all this is that complex societies are intrinsically associated with a non-egalitarian structure. A corollary to this is the need for a degree of centralization or hubs to maintain any degree of complexity beyond the ordinary. We see the same constraint imposed on several biological networks, such as transcriptional networks. Here, there are small number of transcription factors, “hubs”, that control a very large number of genes and a large number of transcription factors that control a small number of genes. In such a network structure, where the degree-distribution of the nodes follows a power-law distribution, an attack that randomly targets nodes has a low probability of breaking up the network. However, an attack that specifically targets the hubs will surely bring it down. However, such networks do fine with regime change, i.e., nodes which are hubs may be freely lost as long as new hubs with statistically similar properties as the lost hubs concomitantly emerge in the network. Hence, we may say that any defect that can propagate up to the hubs so as to irreplaceably destroy them cause a collapse of the system.

In our youth the Rus, who had implemented the Marxian delusion, plied us with cheap books with subtle delivery of their cherished doctrines: 1) Egalitarianism; 2) Communism; 3) triumph of technology over religion and nature. Our meditations on such issues in the social domain were initially sparked by our trying to apprehend the first of these – the superficially alluring egalitarianism of the Marxian doctrine. At that time many around us had bought it into it. At the face of it seemed like channel for rebellion against authority – being a svacchanda we were not fond of authority ourselves. Yet our closest informers had warned us about the dangers of the Marxian doctrine. While playing all these issues in our mind, we soon realized that it was a figment of the Abrahamistically deluded imagination. Thereafter we soon arrived at the interlocking of complexity, structure, functionality and inequality. That interlocking acted like a conservation principle you could increase one side of this without reducing the other – essentially you cannot have your cake and eat it too. Thus, social complexity, intra-social structure and effective multifunctionality went hand in hand with increased inequality. Egalitarianism simply meant that such things had to devolve towards simplification or entirely collapse, since the climb-down is never a mirror image of the climb-up. If the climb-up were to happen by the mode of preferential attachment of new nodes to certain nodes of the network, the climb-down it often proceeds through an attack on hubs of a network with properties approximation a power-law distribution resulting in collapse. In course of our life we have witnessed several examples of such collapse under Marxian or semi-Marxian implementations.