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Many years ago, I was surprised to learn that Andrew Odlyzko does not consider the existing computational evidence for the Riemann hypothesis to be overwhelming. As I understand it, one reason is as follows. Define the Riemann–Siegel theta function by $$\vartheta(t) := -\frac{t}{2}\log\pi + \arg \Gamma\left(\frac{2it+1}{4}\right)$$ and define the Hardy function $Z(t) := e^{i\vartheta(t)} \zeta(1/2 + it)$. Then $Z(t)$ is real when $t$ is real, and the Riemann hypothesis implies that for $t$ sufficiently large, $Z(t)$ has no positive local minimum or negative local maximum, so its zeros are interlaced with its minima and maxima. On the other hand, as I believe Lehmer ("On the roots of the Riemann zeta-function," Acta Mathematica 95 (1956), 291–298) was the first to point out, there exist "Lehmer pairs" of zeros of $Z(t)$ that are unusually close together, which may be regarded as "near counterexamples" to the Riemann hypothesis. Harold Edwards has suggested that Lehmer pairs "must give pause to even the most convinced believer in the Riemann hypothesis."

There is a relationship between Lehmer pairs and large values of $Z(t)$. It is known that $Z(t)$ is unbounded, but it approaches its asymptotic growth rate very slowly. As Odlyzko has explained (see Section 2.9 in particular), there is reason to believe that current computations are not yet exhibiting the true asymptotic behavior of $Z(t)$. So one could argue that the existing computational data about Lehmer pairs is still in the realm of the "law of small numbers."

A related observation concerns $S(t) := \pi^{-1}\arg\zeta(1/2 + it)$. Let me quote from Chapter 22 of John Derbyshire's book Prime Obsession, where among other things he reports on a conversation he had with Odlyzko.

For the entire range for which zeta has so far been studied—which is to say, for arguments on the critical line up to a height of around $10^{23}$—$S$ mainly hovers between $-1$ and $+1$. The largest known value is around 3.2. There are strong reasons to think that if $S$ were ever to get up to around $100$, then the RH might be in trouble. The operative word there is "might"; $S$ attaining a value near $100$ is a necessary condition for the RH to be in trouble, but not a sufficient one.

Could values of the $S$ function ever get that big? Why, yes. As a matter of fact, Atle Selberg proved in 1946 that $S$ is unbounded; that is to say, it will eventually, if you go high enough up the critical line, exceed any number you name! The rate of growth of $S$ is so creepingly slow that the heights involved are beyond imagining; but certainly $S$ will eventually get up to $100$. Just how far would we have to explore up the critical line for $S$ to be that big? Andrew: "Probably around $T$ equals $10^{10^{10,000}}$." Way beyond the range of our current computational abilities, then? "Oh, yes. Way beyond."

In light of what I learned from another MO question of mine, about fake integers for which the Riemann hypothesis fails, I got to wondering—If exploring $Z(t)$ and $S(t)$ for the actual Riemann zeta function is hitting our computational limits, could we perhaps gain some insight by computationally studying other zeta functions? More specifically:

  1. Are there $L$-functions in the Selberg class for which there are analogues of $Z(t)$ and $S(t)$ which are computationally more tractable than the Riemann zeta function, for which we could computationally explore the analogue of the "$S\approx100$" regime? (Incidentally, I don't understand what is significant about the $S\approx100$ regime. Anybody know?)

  2. Are there Beurling generalized number systems for which the analogue of RH fails but which can be shown computationally to mimic the empirically observed behavior of $Z(t)$ and $S(t)$ (including, I guess, the GUE phenomenon)?

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    $\begingroup$ I'm far from being an expert on this subject, but to me the existence of Lehmer pairs looks more like a local phenomenon, while RH is a global one. Really the (multi)set of non trivial zeros of Zeta should be viewed as an entity by itself, just like the Selberg class is to be considered as a meaningful structured object and not an arbitrary collection of moreorless related different functions. $\endgroup$ – Sylvain JULIEN Sep 3 '18 at 20:26
  • $\begingroup$ There is computational restriction and there is representational restriction. If we have to examine 10^80 values of some function to draw an inference/try a particular proof technique, there are space as well as time issues. Are you proposing a compactification or re-presentation in such a way that fewer values need to be checked (never mind the cost of computing these values)? Or is your concern purely computational, which is finding an alternate function which "yields information" using current computational resources? Gerhard "How Big Is Really Big?" Paseman, 2018.09.03. $\endgroup$ – Gerhard Paseman Sep 3 '18 at 20:29
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    $\begingroup$ On the other hand, it is true that RH if true "is only barely so" as it is equivalent to the non positivity of the de Bruijn-Newman constant. Since Tao and Rodgers' breakthrough proving Newman's conjecture, we now know that RH holds true if and only if this constant is null. $\endgroup$ – Sylvain JULIEN Sep 3 '18 at 20:30
  • $\begingroup$ @SylvainJULIEN : I think that what you say in your first comment makes sense if RH is true, but is perhaps less convincing if RH is false... $\endgroup$ – Timothy Chow Sep 3 '18 at 21:00
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    $\begingroup$ Some generalized $L$-functions can have zeroes with multiplicity. Perhaps the most interesting simple examples are the $L$-function of elliptic curves. Under the BSD conjecture, there is a zero at the center of the critical line with nontrivial multiplicity if and only if the rank is at least two. So these all give near counterexamples to the Riemann hypothesis, and many are quite computationally tractable. In some cases it is possible to verify the RH at this point, by existing work towards BSD (Gross-Zagier, Kolyvagin) but I don't think it is otherwise possible to verify computationally. $\endgroup$ – Will Sawin Sep 4 '18 at 17:59
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To put things in perspective, it is not only Andrew Odlyzko who thinks the existing numerical evidence is unconvincing; look at the paper On some reasons for doubting the Riemann Hypothesis by Ivic, or The Riemann Hypothesis by Conrey, or Problems of the Millennium: the Riemann Hypothesis by Sarnak. More or less every expert in the field thinks this way.

In answer to your first question, the $L$-functions which we already know or believe to be in the Selberg class are automorphic $L$-functions, which are no less complicated than the Riemann zeta function computationally.

As Julien pointed out in the comments, the existence of Lehmer pairs for the Riemann zeta function is related to the Newman conjecture for the de Druijn Newman constant, now known to be true by the work of Rodgers and Tao: "the Riemann Hypothesis, if true, is only barely so". The Newman conjecture has a formulated analog for quadratic Dirichlet $L$-functions (google 'Low discriminant') for which Rodgers and Tao expect their proof to generalize. Likely this could be extended further. So the situation is just as delicate for any Selberg class $L$-function, conjecturally.

I expect there is nothing sacred about $\pm100$, just the intuition that $\pm1$ is not large. And $10^{10^{100}}$ is computationally out of range.

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  • $\begingroup$ Thanks for this extra information, especially Ivic's paper. I'm a bit puzzled why you cite Conrey and Sarnak, because Sarnak says that he is an optimist and believes in GRH, while numerical computations are the first thing Conrey cites as evidence for RH. Odlyzko not only finds the computational evidence inconclusive, but is an agnostic about RH itself. It seems to me he is in the minority among number theorists. $\endgroup$ – Timothy Chow Sep 5 '18 at 1:00
  • $\begingroup$ @TimothyChow In my haste I did not choose the best possible examples. But I would argue that when Conrey says 'Billions of zeros can't be wrong', he is presenting the case for RH; not arguing his true beliefs. Sarnak does talk about the (enormous) size of a value of $T$ where a counterexample might be found. $\endgroup$ – Stopple Sep 5 '18 at 6:13
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    $\begingroup$ It is crucial here to make a distinction between belief in RH, and belief that the numerical evidence is convincing. It seems like splitting hairs, but it is a matter of some subtlety. I'll try to find better citations, but meanwhile I'll stick with my assertions that experts understand the numerical evidence is not convincing.. $\endgroup$ – Stopple Sep 5 '18 at 6:16
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Let $p_k$ be the $k$-th prime number, and pick a sequence of primes $q_k$, such that $q_k\sim p_k^{3/2}$. Let $G$ be the arithmetic semigroup consisting of all integers not divisible by one of the $q_k$. Then $\zeta_G(s)=\zeta(s)\prod_{k=1}^\infty(1-q_k^{-s})$. Now $$ \sum_{k=1}^\infty\log(1-q_k^{-s})-\log(1-p_k^{-3s/2}) =\sum_{k=1}^\infty \frac{p_k^{3s/2}-q_k^s}{(p_k^{3/2}q_k)^s}+H(s), $$ where $H$ is uniformly bounded in $\Re s>1/3+\delta$. From the prime number theorem for short intervals we see that we can pick the sequence $q_k$ in the such a way that $|p_k-q_k^{3/2}|<k^{4/5}$, and we find that the series converges for $\Re s>\frac{1}{5}$. We conclude that $\zeta_G(s)=\zeta(s)\zeta(3s/2)^{-1}H(s)$, where $H$ is holomorphic and zero free in $\Re s>\frac{1}{3}$. Assuming RH we find that in the half plane $\Re\;s>\frac{1}{3}$ we have that $\zeta_G$ behaves exactly as $\zeta$, but it has an additional zero at $\frac{2}{3}$. So if you believe in GUE for $\zeta$, then this function satisfies GUE, but not RH. If you define $S(t)$ as the error term in the approximate formula for $N(t)$, then $\zeta_G$ and $\zeta$ share $S(t)$, but $\zeta_G$ does not satisfy RH. However, this example does not say anything about $Z(t)$, because although the function $H(s)$ is holomorphic and zero free, at one point in the computations we took derivatives, thus $H(s)$ might grow exponentially with $|\Im s|$. It is not clear whether you can pick the sequence $q_k$ in such a way that $\zeta_G$ has only polynomial growth.

A more natural example to study would be Selberg $\zeta$-functions. These functions behave pretty much like the Riemann $\zeta$-function. Selberg $\zeta$-functions satisfy RH, with possibly some real exceptional zeroes (and there are cases where these exceptions actually occur). However, in this case $S(t)$ is much larger than in Riemann's case, so this might not be such a good analogue.

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  • $\begingroup$ What would be the best possible abscissa of convergence for the product $H(s)$ ? $\endgroup$ – reuns Sep 8 '18 at 14:37
  • $\begingroup$ @reuns: There are two sources of non-convergence: Truncating the Taylor series of $\log$, and choosing primes close to given points. The first can easily be improved as much as you want. The second is more difficult. If you assume Cramer's conjecture, i.e. $p_n-p_{n+1}\ll\log^2 n$, you can make $p_k^{3/2}-q_k$ as small as $k^\epsilon$, and the series converges for $\Re s>0$. Unconditionally you could improve the bound by using the fact that very large differences between primes are somewhat rare, see Heath-Brown, "The differences between consecutive primes III", but details become difficult. $\endgroup$ – Jan-Christoph Schlage-Puchta Sep 9 '18 at 9:29

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