Simple zeros of primitive Dirichlet $L$-functions and the asymptotic large sieve

TL;DR

Under GRH, the paper demonstrates that 91% of zeros of primitive Dirichlet L-functions are simple using the asymptotic large sieve, improving previous bounds and analyzing pair correlation functions.

Contribution

It provides a higher lower bound on the proportion of simple zeros and computes a q-analogue of the pair correlation function for primitive Dirichlet L-functions.

Findings

91% of zeros are simple under GRH

Computed q-analogue of pair correlation function

Improved previous proportion bounds from 86% to 91%

Abstract

Assuming the Generalized Riemann Hypothesis (GRH), we show using the asymptotic large sieve that 91% of the zeros of primitive Dirichlet $L$-functions are simple. This improves on earlier work of \"{O}zl\"{u}k which gives a proportion of at most 86%. We further compute an $q$-analogue of the Pair Correlation Function $F(\alpha)$ averaged over all primitive Dirichlet $L$-functions in the range $|\alpha| < 2$ . Previously such a result was available only when the average included all the characters $\chi$.