Fatma Cicek | Searcharxiv

arXiv2023

Mean values of long Dirichlet polynomials with divisor coefficients

In this article, we prove an asymptotic formula for the mean value of long smoothed Dirichlet polynomials with divisor coefficients. Our result has a main term that includes all lower order terms and a power saving error term. This is derived from a more general theorem on mean values of long smoothed Dirichlet polynomials that was previously established by the second and third authors. We thus establish a stronger form of a conjecture of Conrey and Gonek in the case of divisor functions.

math.NT↗

arXiv2021

On the Logarithm of the Riemann Zeta-function Near the Nontrivial Zeros

Assuming the Riemann hypothesis and Montgomery's Pair Correlation Conjecture, we investigate the distribution of the sequences $(\log|ζ(ρ+z)|)$ and $(\argζ(ρ+z)).$ Here $ρ=\frac12+iγ$ runs over the nontrivial zeros of the zeta-function, $0<γ\leq T,$ $T$ is a large real number, and $z=u+iv$ is a nonzero complex number of modulus $\ll 1/\log T.$ Our approach proceeds via a study of the integral moments of these sequences. If we let $z$ tend to $0$ and further assume that all the zeros $ρ$ are simple, we can replace the pair correlation conjecture with a weaker spacing hypothesis on the zeros and deduce that the sequence $(\log (|ζ^\prime(ρ)|/\log T))$ has an approximate Gaussian distribution with mean $0$ and variance $\frac12\log\log T.$ This gives an alternative proof of an old result of Hejhal and improves it by providing a rate of convergence to the distribution.

math.NT↗