Analogues of the Riemann zeta function that are more computationally tractable?

Question

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)?

Answer 1

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.

Answer 2

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.