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

necessarycondition for the RH to be in trouble, but not asufficientone.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.Waybeyond."

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:

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?)

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)?