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

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 Özlük 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 $\chi$.

This work was initiated during the Arithmetic Statistics MRC program at Snowbird, Utah. Corollary 3 and Section 7 are added

References in corpus (1)

• Asymptotic Large Sieve