# Lehmer pairs and GUE

In this post we assume the Riemann hypothesis and the simplicity of zeroes, thus the zeroes of $latex {zeta}&fg=000000$ in the critical strip take the form $latex {frac{1}{2} pm i gamma_j}&fg=000000$ for some real number ordinates $latex {0 < gamma_1 < gamma_2 < dots}&fg=000000$. From the Riemann-von Mangoldt formula, one has the asymptotic

$latex displaystyle gamma_n = (1+o(1)) frac{2pi}{log n} n &fg=000000$

as $latex {n rightarrow infty}&fg=000000$; in particular, the spacing $latex {gamma_{n+1} – gamma_n}&fg=000000$ should behave like $latex {frac{2pi}{log n}}&fg=000000$ 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 $latex {gamma_n, gamma_{n+1}}&fg=000000$ such that

$latex displaystyle frac{1}{(gamma_{n+1} – gamma_n)^2} geq 1.3 sum_{m neq n,n+1} frac{1}{(gamma_m – gamma_n)^2} + frac{1}{(gamma_m – gamma_{n+1})^2}. (1)&fg=000000$

The specific value of constant $latex {1.3}&fg=000000$ is not…

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