In this post we assume the Riemann hypothesis and the simplicity of zeroes, thus the zeroes of in the critical strip take the form for some real number ordinates . From the Riemann-von Mangoldt formula, one has the asymptotic

as ; in particular, the spacing should behave like on the average. However, it can happen that some gaps are unusually small compared to other nearby gaps. For the sake of concreteness, let us define a Lehmer pair to be a pair of adjacent ordinates such that

The specific value of constant is not particularly important here; anything larger than would suffice. An example of such a pair would be the classical pair

discovered by Lehmer. It follows easily from the main results of Csordas, Smith, and Varga that if an infinite number of Lehmer pairs (in the above sense) existed, then the de Bruijn-Newman constant is non-negative. This implication is now redundant in view of the unconditional results of this recent paper of myself and Rodgers; however, the question of whether an infinite number of Lehmer pairs exist remain open.

In this post, I sketch an argument that Brad and I came up with (as initially suggested by Odlyzko) the GUE hypothesis implies the existence of infinitely many Lehmer pairs. We argue probabilistically: pick a sufficiently large number , pick at random from to (so that the average gap size is close to ), and prove that the Lehmer pair condition (1) occurs with positive probability.

Introduce the renormalised ordinates for , and let be a small absolute constant (independent of ). It will then suffice to show that

(say) with probability , since the contribution of those outside of can be absorbed by the factor with probability .

As one consequence of the GUE hypothesis, we have with probability . Thus, if , then has density . Applying the Hardy-Littlewood maximal inequality, we see that with probability , we have

which implies in particular that

for all . This implies in particular that

and so it will suffice to show that

(say) with probability .

By the GUE hypothesis (and the fact that is independent of ), it suffices to show that a Dyson sine process , normalised so that is the first positive point in the process, obeys the inequality

with probability . However, if we let be a moderately large constant (and assume small depending on ), one can show using -point correlation functions for the Dyson sine process (and the fact that the Dyson kernel equals to second order at the origin) that

for any natural number , where denotes the number of elements of the process in . For instance, the expression can be written in terms of the three-point correlation function as

which can easily be estimated to be (since in this region), and similarly for the other estimates claimed above.

Since for natural numbers , the quantity is only positive when , we see from the first three estimates that the event that occurs with probability . In particular, by Markov’s inequality we have the conditional probabilities

and thus, if is large enough, and small enough, it will be true with probability that

and

and simultaneously that

for all natural numbers . This implies in particular that

and

for all , which gives (2) for small enough.

Remark 1The above argument needed the GUE hypothesis for correlations up to fourth order (in order to establish (3)). It might be possible to reduce the number of correlations needed, but I do not see how to obtain the claim just using pair correlations only.

## Recent Comments