Saturday, October 26, 2019

Some fun with a proof of the Riemann Hypothesis

A few days ago, Tatenda Kubalalika, a young mathematician in Harare, Zimbabwe, directed my attention to his ambitious paper
Prime numbers and the Riemann hypothesis
To make the story short, I spent about 5 hours over the period of 30 hours before I became sufficiently certain that the paper cannot contain a valid proof of the most prominent open problem of mathematics. Of course I mostly agree with those who say that the Riemann Hypothesis (RH) is very difficult and elementary-looking proofs are almost certainly wrong.

On the other hand, there's no rigorous proof of that assertion – and it's hard to imagine such a rigorous proof of that no-go theorem because you can't even define "elementary-looking". So when someone is excited and passes some tests of not doing something "entirely silly and wrong", I just pay attention. Of course it's been a waste of time so far (dozens of the most famous mathematicians since the 19th century have wasted a lot of their precious time, too) but there's some probability that it could work.

So I am interested not just in my ideas about the RH but also about ideas of others.

If one talks to a relative outsider about some generic topic which may be the RH – but especially about foundational issues in quantum mechanics or quantum gravity etc. – one may almost always discover that the outsider is literally clueless within seconds. This was very different, I needed hours, and decided that even though it's rather clear that the paper can't contain a valid proof of the RH, Tatenda is very intelligent.

I was looking at the most achieved black mathematicians and theoretical physicists. So of course I know folks like Clifford Johnson or Sylvester James Gates. I am not claiming that Tatenda is "at least at the same level". But I believe you will have a very hard time to find comparably good or better young mathematicians in black Africa in 2019. So I do believe that I have interacted with one of the most mathematically talented guys on that (sub)continent.

The alleged proof of the RH has some introduction, Lemmas 1,2,3, and Theorem 4. All these parts deal with some identities relating integrals and sums. The integral is always an integral over \(x\) from \(1\) to \(\infty\) which includes the factor \(1/x^{s+1}\) as well as some difference which is either\[

\pi(x)-\Pi(x), \quad \pi(x)-Li(x), \quad \Pi(x)-Li(x)

\] where \(\pi(x)\) is the prime counting function (number of primes smaller than \(x\), the details don't really matter), \(\Pi(x)\) is its modified cousin, \(\sum_{n=1}^\infty \pi_0(x^{1/n})/n \), and \(Li(x)\) is the estimate of \(\pi(x)\) using prime-independent quantities, namely the indefinite integral of \(1/\log x\) (the detailed fixes don't matter – but his being careful about these details was a reason that made the paper look more trustworthy in my eyes).

The identities are rather straightforward and known to everyone who has dealt with the "translation between \(\zeta\) and prime distributions". The sum on the right hand side is always one of the type \(\sum_{n=2}^\infty\) which is the same summation as one in the definition of \(\Pi(x)\) above except that the \(n=1\) term is omitted.

In Lemma 1, Tatenda proved an identity relating an integral with \(\pi(x)-Li(x)\) in the integrand to a sum – which is almost certainly known. In Lemma 2, he claims that the identity holds for \(Re(s)\gt 1/2\). In Lemma 3, it's said that a similar integral from such identities must converge for a real \(s\gt 1/2\). And because of the convergence, he can sort of show in Theorem 4 that there can't be singularities anywhere in the strip \[

\frac 12 \lt Re(s) \lt 1.

\] That's equivalent to the absence of the zeroes of the zeta function and therefore to the RH.

Meanwhile, aside from dealing with the integrals involving one of the three differences above, he also uses a cute integer approximation of \(Li(x)\), namely \(Li_0(x)\), which is a summation of \(1/\log x\) over integers instead of just an integral. Now, I have sort of seen and worked with all these functions but I wasn't quite certain that I have done all possible clever things with them to know that you can't prove the RH by these operations.

It took hours before I became certain enough that Tatenda was doing something wrong. Well, the first early warning was his "relaxed" attitude to the fact that the logarithm is a multi-valued function in the complex plane – which is why the left hand sides with \(\log(\zeta(s))\) on the left hand side of many equations are really ill-defined in the interesting strip assuming that the RH is violated. He thought that you can just pick a branch and things are fine.

But the choice of the branch of a logarithm is an extremely unnatural move – and the sums or integrals almost certainly never do it for you. They can't create discontinuities around an arbitrary branch. More typically, when there's some singularity like that of \(\log(z)\) for \(z=0\), it means that the sums and integrals will fail to converge in some big region – to make sure that you can't extrapolate the sums or integrals "around" the \(z=0\) singularities of \(\log(z)\) where they would have to pick the "right" branch.

After some time, I saw that he was too "relaxed" about the convergence of all these things. Needless to say, to prove that many of these integrals converge is equivalent to the RH. When you write something like \(\log(\zeta(s))\) as an integral over \(x\), it is clear that the integral will diverge for \[

\frac 12 \lt Re(s) \lt 1

\] if the RH is violated because the integral just must notice that the singularity in the critical strip – coming from \(\log(\zeta(s))\) due to \(\zeta(s)=0\) somewhere in the strip – has the effect on the convergence of the integral. Being sloppy about the regions where the well-defined integrals and sums converge is exactly equivalent to being sloppy about the presence of the RH-violating roots in the strip above i.e. being sloppy about the validity of the RH itself!

And yes, I am now extremely convinced that he's just sloppy. In Lemma 2, he just extends the validity of the identity from Lemma 1. The identity is indeed true if you analytically continue the functions on both sides. But if you don't, there's the problem that the left hand side integral just doesn't converge. And where it diverges depends on the validity of the RH.

For this reason, I suddenly noticed that Lemma 3 was utterly redundant – it just said that some integral converged but the convergence was already implicitly asserted by Lemma 2 because that Lemma said that the integral was equal to something. Well, the most readable warning that something naive and wrong is going on is the beginning of the proof of Lemma 3:
By Lemma 2, note that (6) is an identity and it holds for every real \(s \gt 1/2\). By virtue of being an identity for real \(s \gt 1/2\), it follows that if the right-hand side of (6) converges for real
\(s \gt 1/2\), then so must the corresponding left-hand side.
I sort of understood the statement but believed that he had to mean something else because the rest of the paper looks so technologically refined. But at some moment, the e-mails have assured me that the author really believed the argument above: if an identity exists, then both sides converge either everywhere or in the same region of the complex plane. Well, this assertion is clearly wrong. You may write, to pick a really trivial example,\[

\sum_{n=0}^\infty q^n = \frac{1}{1-q}

\] and it's an identity except that the right hand side is well-defined for any \(q\neq 1\) while the left hand side only convergeces for \(|q|\lt 1\). The same comments apply to integrals that you may evaluate – and to integrals that are equal to a sum by an identity. All these things just generally converge in different regions for the values of the variables! To prove the convergence means to study how the integrands behave in \(n\to\infty\) or \(x\to\infty\) limits etc. You can't prove the convergence by saying that there is an identity (that converges in some region even if it may be continued outside that region in some way).

So his proof of the convergence of \(F(s)\) above \(s\gt 1/2\) is just wrong, based on a naively wrong argument that "the presence of an identity guarantees that both sides converge" – it's an error resulting from a conflation between the "identities between properly evaluated sums and integrals that must converge" and "identities that are obtained from the previous ones by a continuation". So I think that \(F(s)\) converges for the real \(s\gt 1/2\) if and only if the RH holds and he just hasn't proven it one way or another.

The integral converges if you place the "innocent" integrand with \(\pi(x)-\Pi(x)\) inside – because that's equivalent to the removal of the \(\log(\zeta(s))\) singularities from the strip of interest between \(1/2\) and \(1\). However, the other integrands, \(\pi(x)-Li(x)\) or \(\Pi(x)-Li(x)\), simply produce divergence integrals in this strip if the RH is violated and there can't be any "truly trivial" proof of the convergence – and of the RH – based on "look at these well-known identities".

The Riemann Hypothesis really is hard and many simple things of this sort have been tried. The hypothesis may either be formulated as the absence of zeroes of \(\zeta(s)\) in the aforementioned strip; or as "regular enough" distribution of the primes i.e. as the assertion that \(\pi(x)\), the prime-counting function, differs from its non-prime estimate \(Li(x)\) by a difference that is asymptotically smaller than \(x^{1/2+\epsilon}\) for an arbitrarily small \(\epsilon\).

To understand that the RH almost certainly cannot be proven in such elementary steps requires some experience, some mastery of all similar attempts and the reasons why they have to fail. Sociologically, I will adjust my weights – and be more prepared for the scenario that the future authors of proofs of the RH are doing something similarly naive even if the paper makes it clear that they have mastered the basic Riemannian identities and perhaps something on top of them. On the other hand, I don't think that it's "known" that no "basically elementary" proof of the RH may exist. In particular, the claims that the RH is "undecidable" in Gödel's sense seem completely unjustified by anything. And I actually believe that natural well-defined statements about functions such as the RH cannot ever be undecidable.

However, with hindsight, Tatenda's proof is just another example of an attempt that is circular – going between the convergence of sums, convergence of integrals, absence of zeroes in the strip, and nice bounds on the prime-counting function – which can't ever produce a proof because these are just translations between things that are known to be equivalent, and one ultimately makes a logical mistake when he translates too much. ;-)

I have spent hundreds of hours in my life with the RH and most of the "illusions of an imminent proof" have vaporized. So I came to the conclusion that the whole Hilbert-Pólya program is doomed, including various clever stringy variations of it that I had. Right, the set of values of \(t\) for which \(\zeta(1/2+it)=0\) may be the spectrum of a Hermitian operator. The corresponding eigenstates are normalizable. However, even in that map, the RH may simply be false and then the RH-violating zeroes of \(\zeta(s)\) where the real part of \(s\) is between \(1/2\) and \(1\) will correspond to "quasinormal modes" of the Hermitian operator – with the corresponding non-normalizable, divergent eigenstates. And have you eliminated the possibility of such quasinormal modes by finding a nice Hermitian operator with the zeta-like spectrum? I don't think so. There's nothing inconsistent about a quasinormal mode e.g. of a Hermitian operator in some fundamental enough \(AdS_3\) string theory problem or Schnabl-like representations of vacua in string field theory or in some \(p\)-adic string theory or anything you like. You would need an extra argument why that "nice operator knowing about the RH" has no quasinormal modes. What would be such an argument? Some absence of interactions that may be proven differently? How?

So yes, the RH is hard and something seems missing – even in the very general classes of strategies to prove it. But there may still be some easy enough proof. Some index-theorem-like argument or whatever. (Before Michael Atiyah died, I was expecting he would have a link between the RH and index theorems that would be very clever but I was disappointed.)

Meanwhile, I am uncertain whether the lack of responses to Tatenda from the RH-like experts is due to their actual certainty that a proof of this kind cannot exist; or due to their (often hypocritically masked) certainty that a proof of the RH can't possibly come from Zimbabwe; or due to their laziness. My opinion is split but generally, I will keep on looking at ambitious ideas coming from unexpected places – simply because in an overwhelming majority of the cases, I can save the time by immediately seeing that the authors are hopelessly uninformed. That wasn't really the case of Tatenda.

BTW he is on the whitelist here so he can comment here without any moderation – but I can't guarantee that he will be on the whitelist forever. ;-)

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