The Improbable Road to Riemann

Fuji-san on the approach to Narita airport; Debraj Ray (2017)

On June 25, 2021, the Sreenidhi Institute of Science and Technology, located in Hyderabad, released the Report of an "Expert Committee," entitled "Open Reviews of the Proof of the Riemann Hypothesis." The “proof" referred to an unpublished paper by Professor Kumar Eswaran, which claimed to have settled one of the most distinguished open questions in Mathematics, a conjecture stated by Bernhard Riemann in a classic paper written in 1859. As the Preface to the Report states, Eswaran’s contribution “was lying extant on the Internet for nearly four years and had received thousands of reads and downloads but had not received the assent of a mathematical journal to subject it to a detailed peer review." The Committee sought an open review of Eswaran’s paper from 1,200 “eminent mathematicians and scientists." Based on the reports received by the Committee, and its own deliberations, the Committee concluded thus:

“After careful perusal of all the arguments in the proof and the reviews from the experts who have responded, the Committee felt that there are no negative arguments that could technically invalidate the proof and therefore have arrived at the firm conclusion that the proof by Dr. K. Eswaran is both credible and acceptable and that the RH can be considered as proven."

The two-hundred page Report was then released to the Indian press. Reactions in all the major Indian newspapers ran the positive gamut, from enthusiastic corroboration to a sense that an ongoing injustice had been righted. The Hindu, a leading Indian daily, headlined their article with “SNIST physicist finds solution for Riemann Hypothesis." The equally eminent Deccan Chronicle chimed in, “Hyderabad math wizard solves Riemann hypothesis." The Times of India tut-tutted that “there was a reluctance on the part of the editors of international journals to put the paper through a detailed peer review." Shashi Tharoor, Member of Parliament and an acclaimed author with an enormous Indian fanbase (8 million) on Twitter, emitted an exultant tweet:

“The Riemann Hypothesis is a mathematical problem that has remained unsolved for 161 years — until mathematical physicist Kumar Eswaran from Hyderabad did it. It took five years for an expert committee to approve the proof — and a $1 million prize."

We might be admonished for our double-take at Tharoor’s fast and loose mixing of the facts: he is a writer of fiction after all. As it happens, the Committee was set up in January 2020, and delivered its verdict at a pretty fast clip (especially considering the gravity of the claim at hand). And what’s all this about a million dollar prize? Well, there is one, but it isn’t for this Expert Committee to give.

The Riemann Hypothesis is among the list of seven great Mathematics problems, each of which qualifies for a one-million-dollar Millenium Prize of the Clay Institute. Only one of these problems is solved — the Poincaré Conjecture, by Grigori Perelman (who incidentally refused the Prize). The Clay Institute hasn’t printed Eswaran’s check yet, though: its rules state that at least two years must have elapsed since publication, with “general acceptance in the global mathematics community." Alas, with Eswaran’s repeated attempts at publications apparently so unfairly rebuffed, that isn’t about to happen any time soon. Or is it?

Vinayak Eswaran certainly thinks so. He is Kumar’s younger brother, a Professor himself in the Department of Mechanical and Aerospace Engineering at the Indian Institute of Technology in Hyderabad. If you search on YouTube, you will come across his Seven Lectures on the Proof of the Riemann Hypothesis, a silent but well-written self-scrolling document that we found useful to read, and very lucid too. He was invited to provide his assessment to the aforementioned “Expert Committee." In his review, Vinayak writes (emphasis in the original):

I fully believe Kumar Eswaran has proved the Riemann Hypothesis.

As to why he, Kumar, and no other of so many great mathematicians that have chased this Grail, Vinayak’s answer is that Kumar was blessed to find the only key that unlocked the secret door. Invoking both Tolkienesque imagery and Ramanujan’s Goddess of Mathematics, Vinayak writes:

“It is most fortunate that a key to that door even existed, . . . it quite easily may not have, making the [Hypothesis] forever out of reach of proof, driving gifted mathematicians to insanity a thousand years from now . . . I like to imagine that the Goddess looked down from the high heavens and saw a seventy-year-old man reading an abstract tract on the Riemann Hypothesis late into a moonlit night, by the last light of Durin’s day, and said to herself, ‘Why, he loves me truly’, and impetuously slipped him the intuition of the secret door into the Lonely Mountain, a boon she had denied the greatest mathematicians of the 20th Century."

Then there is Kumar Eswaran himself. The title of his hitherto-neglected paper is “The Final and Exhaustive Proof of the Riemann Hypothesis from First Principles," presumably designed to blow-torch all doubt from the minds of errant skeptics. In correspondence with one such skeptic and reviewer for the “Expert Committee," Eswaran writes:

"The Riemann Hypothesis is the greatest open problem in Mathematics. I believe I have solved it comprehensively. I request you to look at the proof again. . . , as I found your review had missed many aspects which I believe, are very crucial to the proof (I beg your pardon for saying so!). This onerous task, which I am asking you to kindly take up, will not only do justice to my work but to the great problem it addresses. . . . In the End, as in all matters of consequence, Truth will prevail."

Eswaran’s paper has been submitted to several journals. From each of these — going by the Expert Committee’s report — the submission has been rejected without peer review. In some cases he has been encouraged to seek the advice of a number theorist or more generally a pure mathematician rather than other engineers or physicists. Apparently, he has tried, without much success. It perhaps does not help either that no member of this Committee of Experts is a pure mathematician.

In this era of Fake News, where even obvious untruths can be proclaimed to serve as Gospel, Kumar Eswaran’s story stands out as irresistibly fascinating. No one reading Kumar’s correspondence with his reviewers can fail to be moved by his sincere, unshakeable belief that he has pulled something off, but that others just cannot see it. No one reading his brother Vinayak’s eloquent prose can fail to be moved by his sense of intellectual excitement and fraternal solidarity. These individuals are not Fake News. They are, as an American might say, the real deal. Which is of course not to say that they are correct, on which more below.

And what about the “Committee of Experts," and their Report? The Report, as we’ve already said, was commissioned by the Sreenidhi Institute, which employs Kumar Eswaran. This Group offers various educational degrees in engineering and charges fees that the market will bear. It is one of several private-sector higher educational institutions which have proliferated in India. Their webpage will cheerily greet you with the message, "Vaccines for All, Free for All! Thank you P.M. Modi!" The Chairman of this Sreenidhi Group is a Dr. K.T. Mahhe, whose biography states that he is an “oft-quoted name in the field of education" who “began his entrepreneurial career in the early nineties, at the young age of 21 years." The high point of his career to date is recorded thus: “Recognition in its truest form came when Sreenidhi became the only private institution among 120 institutions from the region to be selected by the World Bank for assistance." One wonders how that came about, but that story must await a different telling. For the moment, Dr. Mahhe is comfortably (and conveniently) convinced that Kumar has socked it to the Riemann Hypothesis:

“Almost 5 years ago he [Kumar Eswaran] had found a proof of the famous unsolved problem namely ‘The Riemann Hypothesis’. He had put up his proof on the Web and there were very many downloads (numbering in several thousands), and given several lectures on his methods. These lectures were well received and there were no unresolvable negative comments however, in spite of all this there was a reluctance on the part of the Editors of International Journals to put the paper though a detailed Peer review." 

The Chair of the “Expert Committee," Professor P. Narasimha Reddy, followed suit:

“We, as responsible scientists, believe that a paper on such an important problem as RH should not suffer from a lack of review . . . In this connection, more than a thousand invitation letters were sent worldwide by email in the month of February 2020."

Over 1,200 invitations is a more exact number. But this is what the Committee harvested: just three reports (one co-authored). To fill this yawning hole, two committee members then “chose to take upon [sic] this onerous task" of writing additional reviews. A final review was obtained by inviting Professor Vinayak Eswaran, Kumar’s younger brother, on the grounds that he “had taken the trouble to spend the best part of two years to understand and study the background material and understand the proof."

The inside reports are short and sweet. One stated that “the author’s analysis is exhaustive, unambiguous, and every step in the analysis is explained in great detail and established." The other observed that “judiciously using the properties of the random walk problem one shows that Riemann’s Hypothesis is true. There is also a numerical proof given" [emphasis ours, though perhaps an interrobang is called for?!]. In contrast, Vinayak’s invited report was passionate and detailed; we have already quoted from it above, but he has more to say. He notes that Kumar sent his paper to several leading international outlets, but:

“In all cases he received a straight-forward refusal to review the paper, or very cursory review, no more than a paragraph long, that dismissed the paper. Appeals for reconsideration were either refused or just ignored. The reviews invariably gave the impression that the reviewer had not read beyond a few pages, merely looking for a quick reason to reject the paper."

And this brings us to a more sinister turn in the proceedings, one that promises to simmer well in the Fake News vat:

“Contrast this to the way the world mathematics elite treats one of its own. Within living memory, a longstanding great problem in number theory to be solved was Fermat’s Last Theorem. Andrew Wiles, then a professor at Princeton University, proposed a proof in 1993. It was immediately closely examined by other academic mathematicians, and a flaw was found within a few months. Wiles, helped by another mathematician, took a full year to discover a fix for the proof which was finally published in a dedicated issue of the Annals of Mathematics in 1995, and Wiles became a star in the constellation of mathematics genius . . . Kumar’s proposed proof is comprehensible by an undergraduate, and yet, five years on, he still awaits a sincere review of his paper by academic mathematicians!

[There is a] wide prejudice, common in the West but prevalent even in the developing nations, that a breakthrough on a great problem would come only from the West by individuals working at a great western university. Kumar, in other words, is an improbability so remote as to be essentially an impossibility. Which is why, it would seem, no number theorist would even bother reading his paper with any seriousness. Just a different address would have ensured that he got all the help and recognition he needed and deserved five years ago, while now he waits to be formally reviewed for a break-through that would put his name on the front page of every newspaper in the world."

It might help at this stage to say a few words about your authors, Debraj Ray and Rahul Roy. We are two academics with mathematical training. One of us, based in New York, is an economist who works in game theory, the other, based in New Delhi, is a probability theorist. We have been friends since we were children; we have worked together as colleagues. With others and among ourselves, we have wrestled with the themes that Vinayak Eswaran’s anguish brings, distilled and concentrated, seething to the surface. As past Co-Editor of the American Economic Association's American Economic Review, Debraj has constantly been faced with these questions, occasionally (but not infrequently) in an accusatory way. In Rahul’s role as Editor of the Indian National Science Academy's Indian Journal of Pure and Applied Mathematics, these issues crop up, if anything, in reverse: international submissions are just as often rejected as their domestic counterparts. Neither of us personally claims mistreatment at the hands of a discriminatory world; alas, we have been too privileged in our academic upbringing to claim that easy fix for our own professional shortcomings. But the sense around many colleagues and friends that there is insidiously unequal treatment, whether by race, gender or (in this case) international location, is persistent and unceasing. As some of these imbalances receive more corrective attention than others, the others will surely grow.

For this reason, the Eswaran case captivated us. But there were other reasons. Fortunately for our mental health, we do not anticipate solving the Riemann Hypothesis in the near or distant future. That said, we have the basic mathematical knowledge to understand what the Hypothesis is about. We understand the classical roads that have been paved to both its genesis and its solution. We look up to these road builders — and especially to Euler and Riemann — with the greatest of respect. One might liken our knowledge (if stretched optimistically) to a somewhat strained ability to make it, wheezing and puffing, up to Everest Base Camp. The view even at these low altitudes isn’t bad. It enabled us to take a good look at Kumar Eswaran’s paper.

(It goes without saying that the insanely brief and non-technical verbal description that is about to follow can do very little justice to the delicate nature of the problem, but it will have to do for our purposes.)

Briefly, the Riemann Hypothesis concerns locations of the so-called “nontrivial zero" values of a certain distinguished function defined on the plane of complex numbers, called the Riemann zeta function. Specifically, these are the points on the complex plane for which the function returns a value of zero in both its real and imaginary components. The locations of these non-trivial zeros have deep connections with the distribution of prime numbers, a fact that was itself a revelation, connecting as it did the smooth contours of complex analysis with the jagged combinatorial methods of number theory. (And so analytic number theory was born with the presentation of Bernhard Riemann’s classic 1859 paper.) Riemann conjectured that the location of all the non-trivial zeros of his zeta function must lie on a single strip in the complex plane, one corresponding to a real component of 1/2. This is the Hypothesis.

Riemann’s zeta function is only explicitly defined on a proper subset of the complex plane, and then extended to the rest of the complex plane by using the powerful technique of analytic continuation, a method that permits smooth complex functions to be (smoothly and uniquely) continued into areas where they may not be defined by explicit formulae. The insight to finding the zeros of this function comes from “flipping" the zeta function to something closely related: its reciprocal. Just as the original function is defined by analytic continuation, so can the reciprocal function be similarly defined, with an additional feature: where the original zeta function hits a zero, this reciprocal must blow up “to infinity," thus preventing its analytic continuation — which is impossible to do at these blow-up points. Put another way, the failure of the reciprocal function to be analytically continued is then a signal that the function itself is exploding to infinity, which would then signal that Riemann’s zeta function is about to hit zero. Turning that last point on its head, if we can continue the analytic continuation all the way back to the vertical strip identified by Riemann, there cannot be any zeros outside that strip, a finding that would prove the Hypothesis.

This insight comes from the great British mathematician John Edensor  Littlewood, and it is an insight that Eswaran follows in his paper, without additional novelty, until we arrive at the particular point where we must determine whether the analytical continuation of the reciprocal function is possible or not. Extending our Himalayan analogy, this leads us into a wonderfully deceptive little plateau where we can catch our breath. Whether that continuation is possible depends on the property of a sequence of binary numbers, the so-called Liouville λ function, which just takes two values: +1 if its rank in the sequence (ignoring 1) is the product of an even number of prime numbers, and −1 if its rank is the product of an odd number of primes. This strange and beautiful object behaves erratically as we proceed down the sequence: a few +1’s are suddenly followed by a −1, and vice versa, for all intents and purposes “behaving randomly" as the sequence wears on. Such weird behavior by anything that has to do with prime numbers comes as no surprise to any number theorist, of course. Don’t take our word for it: this Wikipedia link will introduce you nicely to Liouville, has lots of diagrams, and will also show you how the function is connected to the reciprocal form of the Riemann zeta function, in exactly the way that Eswaran correctly makes that connection. Indeed, this connection has been made already in the work of Edmund Landau, and there is a substantial literature on it; see, for instance, Borwein, Ferguson and Mossinghoff (2008) and a more recent expository article by Osler and Wright (2019).

But it is at this point, alas, with Everest tantalizingly visible on the horizon (thanks to John Littlewood), that Kumar Eswaran makes his mistake.

The fact that the negative and positive values of the Liouville λ function looks random does not allow you to conclude that it is random. The λ function is profoundly deterministic: for each rank as we march down the ordered list of natural numbers, there is not the slightest smidgen of a doubt regarding its unique prime number factorization, and therefore about the number of primes that enter into its factorization. In particular, whether that number is even or odd is fixed since and for all eternity. We must tread very carefully on this slippery terrain, and credit must be given to Eswaran where it is due. We do not intend to suggest that Eswaran views λ as random. His error — or rather his leap of faith — is to imagine that the limit theorems pertaining to truly random variables can be applied to this deterministic sequence, simply because they have the aura of pseudo-randomness. These limit theorems are elementary — any advanced Math major knows them — which explains Vinayak’s exasperated exclamation above that “Kumar’s proposed proof is comprehensible by an undergraduate"! Yes, it is comprehensible, but correctness is another matter altogether. To make proper headway, one would need to embed λ into a stochastic process defined on the space of binary-valued sequences, and then slay the demon of whether the particular sequence of Liouville is “generic" in that probability space, so that limit theorems (which are only probability-one statements, after all, not statements about every sample path) can be brought to bear. 

Oh, it is still a long way up. Or worse still, we have redefined our vista with Littlewood’s help, but the new problem at hand has all the diabolical subtleties of the Riemann Hypothesis nestled within it. It is a restatement of the Hypothesis, if you will, not its resolution.

We will rest our own examination there, and turn to the three outside reviewers. By and large, their response isn’t different, which comes as no surprise to us. All the reviews are courteous and engage fully with the paper, but they raise doubts. The co-authored review (by Ken Roberts and S. R. Valluri of the Physics and Astronomy Department, University of Western Ontario) makes several points, and among them notes:

“It is worthwhile to give more thought to the concept of how a deterministic sequence might be ‘sufficiently like’ a random walk. At various places in the text . . . , the assertion is made that any sequence (deterministic or not) which shares certain properties with unbiased random walks will necessarily satisfy [standard limit theorems]. We are not confident of that assertion. It requires further justification, in our opinion. It would be a useful result, and deserves to be explored fully."

They also observe that some aspects of Kumar Eswaran’s exertions in this regard could lead to separate byproducts of interest, and these “should be published so they are made known to a wider community of scholars."

A second report by Germán Sierra (of the Institute of Theoretical Physics in Madrid) is more dismissive:

“[T]he apparent random walk nature of the Liouville function is a well known fact in analytic number theory. It is heuristic and, according to most expectations, unlikely to be true. The reason being that the prime numbers are deterministic objects, not random, as well as the Liouville and other arithmetic functions. These objects seem to behave randomly but they do not. Their apparent random nature has been very useful in the past to propose ‘conjectures’ but it is impossible to use them to ‘prove’ anything about the primes or related quantities."

The third report is the only report from a mathematician: a distinguished Polish number theorist. His name is Wladyslaw Narkiewicz. He is 85 years old. His email correspondence with Kumar Eswaran is nothing short of moving. He makes the same points again and again, connected to the arguments we have already given, so there is little point in repeating them. What is touching is the gentleness which imbues his exchanges with Kumar, reiterating his points patiently but firmly. He seems to feel Kumar’s desperation, his need to be right, and one cannot read this exchange (on both sides) without being emotionally affected:

“I looked again thoroughly at your text and your explications. I subsumed my thoughts about your paper in the attached file. Unfortunately there is a deep hole in your argumentation, when you assert without any proof that a theorem in the theory of random sequences implies a bound for the sums of values of the function λ(n) . . . If you really have a proof of your assertion, then I would like to be able to see it.

Do not worry about this situation. Several excellent mathematicians tried without success to prove Riemann Hypothesis. With every good wish."

Those last lines might look like a sarcastic putdown, but in the overall context of the correspondence, they are not. Setting that aside, Kumar Eswaran’s reply is illuminating (edited very lightly for obvious typos):

“As far as I can tell, I have responded to almost all your queries. [Our note: he has not.] Now I wish to explain, in my own way of thinking, my surmise that the RH is true. In order that you will appreciate what I say I now request you to think from my point of view i.e. the point of view of a Theoretical Physicist which is somewhat different from the point of view of a strict Number Theorist as I will presently explain. I request you to read the rest of this email and try to empathise with my point of view.

In mathematics a Mathematician starts from some initial premise. . . a geometer starts from (say) Euclid’s Axioms and a Number theorist starts from say Peano’s axioms and the axioms of logic, [and] with these as a basis the Geometrician/Mathematician discovers new theorems which are essentially logical deductions from the Axioms. But a Theoretical Physicist starts from the study of a phenomenon as observed in nature or as viewed in an experiment. He then tries to formulate ‘laws’ which can explain these phenomena. Of course there are limitations in both viewpoints. The Mathematician can never discover anything that lies beyond the reach of his axioms (this aspect has been spectacularly demonstrated by Godel by his Incompleteness Theorem) and the Physicist can never discover any “Law" without having had the opportunity to view the phenomena or conduct an experiment."

It may be best to refrain from comment here. Suffice it to note that Professors Narkiewicz and Eswaran part on amicable terms. Here is an extract from the mathematician’s last email to Kumar:

“Although our views on your result are somewhat different I want you to know that I enjoyed our discussion which showed that the notion of a proof may have different interpretations. I want also to stress that the word ‘heuristic’ has no negative meaning. A lot of work of really great mathematicians has been performed in a heuristic way. This applies not only to old times (Euler, Laplace, the Bernoullis, . . . ) but also to recent times.

Because of the pandemic I sit at home since three weeks, but this permits me to read all the books which I bought a time ago and did not have time to read them earlier. I am trying also to do some mathematics but my age limits my possibilities."

Ah Professor Narkiewicz, you speak for many more of us than you know.

So if we set aside the machinations of the “Expert Committee" on Kumar Eswaran’s purported proof of the Hypothesis, there is another story here, which deserves a separate telling: what constitutes proof ? That rather gracious concession by Professor Narkiewicz might enliven the hearts of many a physicist (and certainly many an economist!), but Gödelian uncertainties notwithstanding, your authors remain willingly and firmly marooned on the shores of mathematical logic.

In our reading of this entire debate, some concluding remarks come to mind. First, has Kumar Eswaran’s work really been ignored by 1,196+ mathematicians? We think not. The error is there even for us to see, and there is absolutely no doubt in our minds that any mathematician would see it immediately. They might be too lazy, or timid, or time-consumed to respond, simply to point out obvious lacunae. This entire study constitutes one data point, but a cautionary one. There may be statistical discrimination in academia, whereby a submitted paper from Hyderabad or La Paz or Beijing is given short shrift. But one needs to understand such statistical discrimination within the contours of a world that is absolutely jam-packed with the need to process information, and to do so quickly. It is possible, and especially so in the more imprecise disciplines such as Economics, that statistical discrimination is over-used. But this case, to us, is emphatically not a case in point. It remains an open question, though, as to whether all intellectual genius must inevitably reveal itself.

Second, Bernhard Riemann’s conjecture deserves the deepest respect, as do the many deep problems in mathematics, philosophy and logic that confront us, and that we often brush aside with a flippant wave of our smartphones (which ironically owe their existence to the solutions of some of these problems). This is not a political discussion, or a question of the future of the Sreenidhi Institute’s fund-raising or fee-charging capabilities. It is appalling to see how the “Expert Committee" reacted to the fact that only 4 of their 1200+ stepped forward. Is there not something to be gleaned from that last observation? Might it be the case that over 1000 eminent mathematicians would simply ignore a deeply structured attempted proof of one of the greatest conjectures in Mathematics? Might there not be something to be inferred from this stony refusal, or should we rather believe that there is a grand conspiracy of silence directed against scientists from the developing world? Dr. Mahhe, the entrepreneurial Chairman of the Sreenidhi Group, whom we have heard from before, is asking to be heard again:

“[T]he status of the Riemann Hypothesis in mathematics is so great that very few were willing to take the risk of openly venturing an opinion on the correctness or the incorrectness of the proof. Only seven scholars responded. The author of the proof responded to these comments, where warranted."

Words fail us in seeking an adequate response to this particular inference, except to observe that it is a truly entrepreneurial one.

And finally, there is the question of the three outside reports that were, in fact, sent to the Committee. What of the many doubts and queries that were raised by four individuals in their three painstaking reports? In concluding this article, we can only let the Committee speak for itself, repeating and extending the quote that we started with:

"After careful perusal of all the arguments in the proof and the reviews from the experts who have responded, the Committee felt that there are no negative arguments that could technically invalidate the proof and therefore have arrived at the firm conclusion that the proof by Dr. K. Eswaran is both credible and acceptable and that the RH can be considered as proven.

The Committee recommends that the detailed unexpurgated Reviews along with the comments of K. Eswaran and his papers and the findings of this Expert Committee, be compiled in the form of an e-book with a suitable Preface, be made available to the world’s scientific community for their perusal and for the historical record."

We leave you, dear Reader, to judge whether the “firm conclusion" of the Committee is justified.

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(This is the first draft of an article with Rahul Roy, Professor at the Indian Statistical Institute in New Delhi. Rahul has kindly allowed me to publish this first iteration while he works on the second. We will be updating soon.)