# Lehmer pairs and GUE What's new

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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## 3 thoughts on “Lehmer pairs and GUE”

1. Brian says:

Is there a third option crisply different and distinct from Boolean logic’s answer to P/NP

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1. Brian says:

And, if so, could that open the door for taking Skolem’s ternary logic paradigm more respected and perhaps the proper “paradigm” (which I loosely define as Logic+Set.theory) to prove that PNP must be stated and potentially proved in the aforementioned paradigm?

Skolem wrote this in the 1930’s or so but memory doesn’t serve me well!

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