An unconditional Montgomery theorem for pair correlation of zeros of the Riemann zeta-function (Q6595593)

Let \(\rho = \beta + i\gamma\) denote a nontrivial zero of the Riemann zeta-function \(\zeta(s)\) with \(\beta, \gamma \in \mathbb{R}\), that is, a zero satisfying \(\beta> 0\). The Montgomery pair correlation method relies on the function \(F(x,T)\) defined for \(x>0\) and \(T\geq 3\) by \N\[\NF(x,T)=\sum_{\stackrel{\rho,\, \rho'}{0<\gamma,\, \gamma' \leq T}}x^{\rho-\rho'}W(\rho-\rho'), \N\]\Nwhere \(W(u)=\frac{4}{4-u^2}\). The function \(F(x,T)\) can be normalized by defining for \(\alpha>0\) \N\[\NF(\alpha)= \left(\frac{T}{2\pi}\log T \right)^{-1} F(T^\alpha, T).\N\]\NThe main result of this paper is that unconditionally the function \(F(\alpha)\) is real, even, and nonnegative. Moreover, as \(T \to \infty\), we have \N\[\NF(\alpha)=(T^{-2\alpha}\log T +\alpha)\left(1+O\left(\frac{1}{\sqrt{\log T}}\right) \right) \N\]\Nuniformly for \(0\leq \alpha \leq 1\).\N\NAs consequence, if we suppose that all the zeros \(\rho=\beta+ i\gamma\) of the Riemann zeta-function with \(T^{3/8}< \gamma \leq T\) lie within the thin box \N\[\N\frac{1}{2}-\frac{1}{2\log T}<\beta <\frac{1}{2}+\frac{1}{2\log T} \N\]\Nthen for any sufficiently large \(T > 0\), at least \(61.7\%\) of the nontrivial zeros are simple.\N\NNote that the pair correlation method developed in this paper neither requires nor provides any information as to whether or not the nontrivial zeros of \(\zeta(s)\) satisfy \(\beta=1/2\).