Monday, November 23, 2015

Could have a Nigerian teacher proved the Riemann hypothesis?

Not really, LOL, thanks for asking

The Clay Institute has announced its "seven millennium problems", a group of deep mathematical conjectures that had been neither proven nor disproven. To solve any of them means to win $1 million. Famously enough, Grigory Perelman has made the crucial steps in proving the Poincaré conjecture – a statement of the kind that "if something quacks and smells like a three-sphere, it is a three-sphere" – and rejected the bounty.

All six other problems remain unsolved so no money has been paid at all. I am among those who consider the Riemann Hypothesis to be the most profound conjecture among the seven. To say the least, I have spent much more time with efforts to prove it (and yes, I mostly believe it is true) than with the other six combined.

A week ago, all leading British media have brought us wonderful news: a Nigerian teacher named Opeyemi Enoch has solved the problem and will be paid $1 million.

The U.K. journalists were in a complete consensus about that, regardless of the left-right divisions. The BBC recorded an audio interview and asked what he would do with $1 million. The Daily Mail has announced that the money has actually been already paid! The Telegraph and The Independent were among those celebrating, too.

Holy cow. It would be great if a guy like that from the Nigerian countryside – not even the capital – solved the greatest problem of mathematics. In fact, it would be so great that it's downright ludicrous. Try not to lose your common sense altogether.

According to many surveys, Nigeria is the world's 7th lowest-IQ country with the average IQ at 67. (Some of the surveys seem to claim numbers up to 84 but they seem to be exceptions.) If you agree that the IQ above 160, four standard deviations above the mankind's average, is basically a necessary condition for solving the problem, then Nigeria needs more than 6 standard deviations to get to 160. An excess higher than 6 sigma occurs in "1 case in 1 billion". But there are many fewer people than 1 billion living in Nigeria.

If you receive an e-mail from Nigeria that promises you to be paid $20 million, it's possible that someone rich in Nigeria is really trying to share his wealth with you. Alternatively, it's a scam – the so-called Nigerian e-mail scam. I guess that most people in the West have understood that the second possibility is far more likely. The case of the Riemann hypothesis is analogous.

We don't need to talk about IQs. You may also say that a groundbreaking proof of the Clay caliber occurs once in 100,000 papers (or whatever is the right number) that are publishable in peer-reviewed journals, or one per 10 million (or another number of) citations. Nigeria isn't producing much of either – papers or citations – so it seems very unlikely that it could produce a groundbreaking result like a proof of the Riemann Hypothesis. In fact, the proportionality that I have referred to heavily overestimates the odds that underdog places achieve something like that. For increasingly more special achievements, the gap between the chances of the top places and the underdogs becomes even more visible than for "regular OK papers".

A sensible article about the tricks used by the Nigerian guy who fooled the British media was written by the journalists in Zambia. Well, sometimes, British readers have to look for Zambian and not British journalists if they need a writer who is smarter than a pumpkin! ;-) One must realize that many British journalists only pretend to be dumber than a pumpkin because it's politically correct and fashionable to do so. (I talk about the "IQ of a pumpkin" because in Czech, it's a popular rhyme: "IQ tykve".) The Zambian article also makes fun of a claim from the BBC audio interview: Enoch's students wanted their guru to "make $1 million off the Internet". Their only relationship to the Riemann Hypothesis is that they saw that one could get some money for that. Yes, I surely believe that the value of a proof of the Riemann Hypothesis would vastly exceed $1 million. Every other idiot may have $1 million but not even the history's best mathematicians could have proven the conjecture.

An even nicer article debunking this ludicrous story was published in the Aperiodical. We learn that the proof of the Riemann Hypothesis wasn't the first big achievement by Prof Enoch:
Dr. Enoch had previously designed a Prototype of a silo for peasant farmers and also discovered a scientific technique for detecting and tracking someone on an evil mission.
I was really laughing out loud when I read this for the first time. You know, these "evil mission" and "silo" contributions are appropriate for someone with IQ around 80, not 160. But they may have a much higher entertainment value.

Well, I've studied the papers and any other sources of information provided by Prof Enoch. There aren't any papers by him. There is just an abstract of a talk he was supposed to give somewhere, plus collected texts that other (more real) mathematicians have written about the problem. The abstract is much more promising – it tries to pursue the strategy of finding a Hermitian matrix/operator whose eigenvalues are the imaginary parts of the Riemann zeta function's zeros. I am pretty sure that this Hilbert-Pólya strategy is considered the "single most promising one" among the successful research mathematicians in the West. So I was impressed that Prof Enoch could have figured it out. But at the end, the documents available at the Clay Institute website are so nicely selected that it shouldn't be shocking that everyone may end up repeating those wisely sounding things. (You may want to read these enlightening quotes and bonus ones about the related mathematical problems.)

I won't bother you with the texts associated with Prof Enoch because there's nothing new over there – and everything that is somewhat good over there has clearly been plagiarized.

But because of this ludicrous Nigerian story, I've spent about 10 more hours thinking about the Riemann Hypothesis since Thursday. I haven't solved it yet but I reminded myself about all my basic strategies that I have pursued over years. Some of those strategies look "more obvious to me" than some years ago – to the extent that I would find it easier to make another step now. Except that I still don't know what's the right way to make another step.

Riemann Hypothesis: basics and my attempts

The Riemann Hypothesis may be expressed in many different ways. One of them is all about the distribution of primes – and looks like pure number theory or arithmetics. Other formulations known to be equivalent are talking about properties of functions of complex variables and seem to be propositions about the nice and smooth calculus.

The possibility to rephrase the claim about primes in terms of functions of a complex variable make the Riemann Hypothesis close to physics. For this reason, some people – which basically includes myself – believe that the Riemann Hypothesis may probably be proven by considering a well-chosen system in mathematical physics (probably quantum physics). Other mathematicians believe that this physicalization will never help – that the problem will ultimately remain a number-theoretical one about the distribution of primes and whenever you embed the Riemann Hypothesis to any calculus- or physics-based system, you are changing nothing about the essence of the problem that is embedded as a whole, too.

OK, what does the hypothesis say? Let me start with the simplest "complex calculus" formulation. Define the Riemann zeta function as\[

\zeta(s) =\sum_{n=1}^\infty\frac{1}{n^s}

\] which converges if the real part of \(s\) is greater than one. For all other values of \(s\in\CC\), it is not hard to see that one may define the function in a unique way so that the function is meromorphic – holomorphic almost everywhere. The simple pole at \(s=1\) is the only singularity of the function; the function is infinite there.

What's harder are the zeroes of the zeta function, i.e. values of \(s\) for which \(\zeta(s)=0\). One may prove that the negative odd numbers \(s=-2,-4,-6,\dots \) are zeroes, the so-called trivial zeroes. And then there are zeroes away from the real axis, the "nontrivial zeroes". All of them may be proven to sit in the critical strip (the real part of \(s\) is between zero and one). Because of a symmetric relationship for \(\zeta(s)\), for a zero \(s\), the numbers \(s^*,1-s,1-s^*\) are zeroes, too.

The Riemann Hypothesis says that all nontrivial zeroes of \(\zeta(s)\) have the real part of \(s\) equal to \(1/2\) exactly.

So they're only paired with \(s^*\) because \(s^*=1-s\) for them. OK, what does this version of the Riemann Hypothesis have to do with the primes? The point is that you may rewrite the original "sum" defining \(\zeta(s)\) as an infinite product, the Euler product:\[

\sum_{n=1}^\infty\frac{1}{n^s} = \prod_{p \text{ prime}} \frac{1}{1-p^{-s}}

\] Why is it so? It's because every \(n\) in the original sum may be written as a product of powers of primes,\[

{\Large n = 2^{e_2} 3^{e_3} 5^{e_5}\dots}

\] where the exponents \(e_2,e_3,e_5,\dots\) are non-negative integers. Up to finitely many exceptions, all these \(e_p\) are equal to zero. The summation over \(n\) may be rewritten as a summation over \(e_2,e_3,e_5,\dots\). The summand is clearly\[

{\Large \frac{1}{n^s} = \frac{1}{2^{se_2}3^{se_3}\dots}}

\] and all the sums that depend on "2" or "3" or "5" or "7" etc. may be separated from each other. Each of these factors ends up being a simple geometric sum and the result of the geometric sum is written in the Euler product formula.

Now, the Euler product formula and its continuations etc. converge or diverge depending on the distribution of primes. What is the probability that a large number comparable to \(N\) is prime? Note that the larger \(N\) is, the harder it is for \(N\) to be prime because there is a greater number of candidate divisors – all numbers between \(2\) and \(\sqrt{N}\), if you wish.

One may show that the probability that a number comparable to \(N\) is prime decreases as \(1/\log N\). Yes, it's a natural logarithm. So if you have numbers around 1 trillion which is \(10^{12}\approx e^{12\times 2.3}\) or so, the probability that it's prime is around \(1/(12\times 2.3)\approx 1/28\).

Needless to say, the primes are not "equally spaced". It's rather hard to decide which large numbers are primes and which are not. Up to the decreasing changes that \(N\) is a prime if \(N\) gets larger, the primes seem to be a relatively "random" set. The Riemann Hypothesis may be shown to be a claim that morally says that "the set of primes doesn't differ much from a random subset of the integers with the decreasing probability". In other words, the Riemann Hypothesis says that there is no "big conspiracy" that would guarantee some "exceedingly unlikely clusters" or "surprisingly huge voids" that would systematically reappear as you go towards larger numbers.

Back to the analytic version of the hypothesis

OK, the function \(\zeta(s)\) has the trivial zeroes at the real axis and, as you may calculate, the nontrivial zeroes of the form\[

s=\frac 12 + it

\] i.e. numbers along the "critical axis" where the imaginary part of \(s\), i.e. \(t\), may have both signs (a trivial symmetry from the complex conjugation) and\[

\pm t \approx 14.13,\, 21.02,\, 25.01,\, 30.42,\, 32.93, \dots

\] These are seemingly irrational – and seemingly transcendental and totally unknown – "random" numbers. The first root has the imaginary part over fourteen; it is pretty far from the real axis. The roots get more frequent as you increase \(t\), the imaginary part. In fact, the density of the zeroes per unit \(t\) becomes, for large \(t\), approximated by \(\log(t)/(2\pi)\). Again, it's the natural logarithm.

This is pretty cool because you should remember that the density of primes had the logarithm in the denominator. Here it is in the numerator, with the extra factor of \(1/(2\pi)\). The similarity is cool because in some sense, the primes and the zeta zeroes seem to behave in the "complementary" ways, like the dual lattices or the momentum-position complementarity or something of the sort. For example, for lattices of electric and magnetic charges, you also get the "inverse spacings" with the extra factor of \(1/(2\pi)\).

I've spent quite some time in my life by attempts to make this duality between "zeroes of zeta" and the "primes" well-defined and rigorous. Needless to say, the simplest models of the "dual lattices" and "phase spaces" don't involve any zeta-function. Lattices don't involve any non-equally-spaced distributions at all. So it's hard. But I am inclined to believe that this idea may be made rigorous and someone – not necessarily me – will succeed in making it well-defined.

If you could show that in some well-defined sense, the set of zeroes of the zeta function is the "dual irregular lattice" to the "irregular lattice of primes", the proof of the Riemann Hypothesis could be as simple as noticing that the primes are real.

The other strategy I've spent lots of time with is the usual one – the Hilbert-Pólya program. Just find a matrix, an \(\infty\times \infty\) matrix, whose eigenvalues are the allowed values of \(t\) listed above. Because \(t\) and \(-t\) are allowed at the same moment, the right matrix may probably be an antisymmetric real matrix. It may be a matrix with a simple form with respect to some basis that represents integers or primes etc. But I don't know what the matrix entries should be.

But the matrix should look random. It's another well-known fact that the zeroes of the zeta function aren't distributed like the Poisson distribution. It's the distribution of random events – like decays of radioactive nuclei in a big piece of matter – in which each moment is uncorrelated to another moment. According to the Poisson distribution, it often happens that three (or even more) events occur very quickly after one another (each other).

But the zeroes of the zeta function seem to be much less likely to be very close to others. They seem to "repel" each other – they are something in between the Poisson distribution and the equally spaced spectrum. In fact, one may see that the statistical distribution seems identical to that of "random matrix theory". Take a random unitary matrix \(N\times N\) with a large \(N\) and compute the eigenvalues. They will be numbers on the unit circle and they will also be unlikely to be close to each other. Their distribution will look statistically identical to that of the correctly scaled Riemann zeta function's zeroes.

So there could exist a single "almost random" matrix whose eigenvalues are simply the numbers \(t\) labeling the Riemann zeta's zeroes. If you found such a matrix and proved that the eigenvalues are the zeroes, then the proof of the Riemann Hypothesis would be equivalent to the proof of the Hermiticity (or anti-Hermiticity, and after the \(1/2\) constant is subtracted) of your matrix which could be straightforward. The "right matrix" could look random but it could be a very specific one – perhaps one equally derived from the distribution of primes or their relative properties. But which ones?

Representations of \(SL(2,\RR)\)

The Hilbert-Pólya program is respected by good mathematicians but it's been around for a century. I am an even bigger fan of a sub-program or modified program of this sort, one that I at least partly invented because I am still not familiar with papers that would formulate this program as clearly as I will do in a minute.

The group \(SL(2,\RR)\) is a continuation of the well-known group \(SU(2)\). Unlike \(SU(2)\) which only has the finite-dimensional irreducible representations, \(SL(2,\RR)\) has five different types of unitary irreducible representations. All of them may carry a label \(\mu\in\CC\) that is analogous to \(m_{\rm min}\) for \(SU(2)\) and the unitarity requires that \(\mu\) is either real or the real part is \(1/2\). This is just like the Riemann Hypothesis conditions for \(s\).

In fact, there's one simple way to unify the real axis with the critical axis \(s=1/2+it\): those are two of the places for which \(s(s-1)\in \RR\). And this \(s(s-1)\) seems to be nothing else than the Casimir of a sort for the representation of \(SL(2,\RR)\).

To get to the point, you could prove the Riemann Hypothesis if you were able to build a nice unitary irreducible representation of \(SL(2,\RR)\) of a special kind for every complex number \(s\) obeying \(\zeta(s)=0\). The construction you would find would have to depend on \(\zeta(s)=0\), perhaps to make your matrices or vectors convergent or normalizable or something like that. Maybe, those representations would form a basis of some "hyperbolic harmonics" on the \(AdS_2\) space. We probably mean the decompactified one, but maybe only some special (rational-number-given) boundary conditions are allowed etc.

More direct links to string theory

At the end, I think that if someone will find a proof of the Riemann Hypothesis at all, it is rather likely that it will be tightly connected with string theory. After all, \(SL(2,\RR)\) mentioned a minute ago is the conformal isometry group of the half-plane or the unit disk. It appears as a subgroup of the Virasoro algebra. The zeta function itself emerges in \(p\)-adic string theory which could provide us with a way to prove the Riemann Hypothesis.

And finally, I've been extremely enthusiastic about proving the conjecture using Martin-Schnabl-style tachyon minimum solution for the cubic string field theory. That solution is a mathematically complicated one and may be written in terms of the zeta function (although for Martin, this seems to be just a bureaucratic trick to reparameterize the Bernoulli number-valued coefficients – the Bernoulli numbers are clearly closely related to the values of the zeta function at integer values of \(s\)).

That approach of mine believes that the absence of the nontrivial zeroes away from the critical axis is equivalent to the disappearance of all physical excitations around the tachyonic vacuum. For this picture to work, one must show that in Martin's system or a very closely related system, it's possible to build a physical excitation around the tachyonic minimum for every \(s\) obeying \(\zeta(s)=0\) – because the equations of motion for the excitations might be basically \(\zeta(L_0)\Psi=0\) in a certain sector and you may choose the eigenvalue of \(L_0\) for your string field \(\Psi\). But for the zeroes on the real or critical axis, these would-be excitations must be unphysical (pure gauge?). This direct way to look for the spectrum could be equivalent to other ways to prove the emptiness of the spectrum of excitations around the tachyonic minimum which could complete the rigorous proof of the Riemann Hypothesis.

I've repeatedly thought that I've been very close to completing a proof of this kind but with the hindsight, I think that all these cases of optimism have been due to my sloppy thinking so far. But maybe someone else will share my excitement about "something hiding in these ideas" and she (perhaps a 9-year-old girl in Uganda) will complete the proof.

If you have missed "String Theory for Kids" by the 9-year-old Peo Webster (who is on "Team String Theory", as she calls it), you should watch it here. She teaches string theory by analyzing Tim Blais' song, Bohemian Gravity. You will learn lots of things, kids, including the differences between AdS/CFT examples for the compact geometries involving \(S^5\) and \({\mathcal T}^* S^3\). ;-)

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