Heuristic model for Lehmer pairs?

Question

Rodgers and Tao proved that the De Bruijn–Newman constant $\Lambda$ is non-negative. The study of $\Lambda$ goes back at least to Lehmer's paper, On the roots of the Riemann zeta-function, whose Figure 1 illustrates the striking phenomenon that the Riemann hypothesis sometimes "looks close to being violated," in the sense that if the curve in Figure 1 were to turn around before crossing the horizontal axis, then RH would fail, and it visually appears to be "dangerously close" to doing so.

I am wondering if there is a plausible heuristic model for Lehmer pairs which treats them as random variables, and which predicts the "probability of a violation of RH" in the above sense. Such a model might give us heuristic confidence in RH, if it lets us say something along the following lines: "If the process really were random, then we would expect a violation of RH by the time the imaginary part of $\zeta(s)$ reached such-and-such a value, but we have calculated the zeros out beyond that without observing a violation. Therefore there must be a ‘reason’ that the violations are not occurring."

A related MO question is Heuristic argument for the Riemann Hypothesis but the above question does not seem to be addressed there.

Answer 1

Assume RH and let $\rho_j = \frac12 + i \gamma_j$ be the $j$th critical zero with positive imaginary part. The number of such zeros with imaginary part in $[0,T]$ is $N(T) = \frac{1}{2\pi} T\log(T/(2\pi e)) + O(\log T)$.

Thus, the "unfolded zeros" $$ \tilde{\gamma}_j := \frac{1}{2\pi} \gamma_j \log(\gamma_j) $$

satisfy: $\tilde{\gamma}_{j+1} - \tilde{\gamma}_j$ is 1 on average. The "GUE Hypothesis" implies that, in the limit as $T\to\infty$, the distribution of $\tilde{\gamma}_{j+1} - \tilde{\gamma}_j$ is the same as the distribution (in the limit $N\to\infty$) of the (similarly rescaled) eigenvalue neighbor gaps of random matrices from the Gaussian Unitary Ensemble of $N\times N$ matrices. Numerical calculations of Ozlyzko support that conjecture.

The PDF of the GUE nearest neighbor distribution is well-approximated by the "Wigner surmise" $\frac{32}{\pi^2} x^2 e^{-\frac{4}{\pi}x^2}$. Thus, at least conjecturally, we know exactly how often there should be extremely small gaps between zeros of the zeta function.

The official definition of "Lehmer pair" quantifies how small the gap has to be, and you can use the GUE Hypothesis to predict the frequency of Lehmer pairs. But in terms of justifying the statement "RH is barely true", it strikes me as sufficient to note that the distribution of neighbor gaps is (conjecturally) supported on all of $(0,\infty)$.

To address what appears to be the point of the original question, note that "barely true" does not mean "likely to be false". The random eigenvalues from the GUE are all real. So, the analogue of RH is known to be true in that context, and yet it also is "barely true". And, the process is random, but not random in a way that suggests RH might fail.

Answer 2

I think the difficulty lies in having a precise definition of Lehmer pair. Lehmer did not make one, but merely exhibited an example of a pair of consecutive zeros with a very small gap, such that the maximum of the Hardy function $Z(T)$ in between was very close to zero. (If $Z(t)$ has a negative local max or positive local min, then RH is false.). Csordas, Smith, and Varga, in Lehmer pairs of zeros, the de Bruijn-Newman constant, and the Riemann Hypothesis, did give a precise definition, and showed that a Lehmer pair gives a lower bound on $\Lambda$. Their definition is somewhat technical, and depends not just on the zeros being close but also on what other nearby zeros are doing.