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

Vorrapan Chandee; Yoonbok Lee; Sheng-chi Liu; Maksym Radziwiłł

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

Number Theory 2013-02-15 v2

Authors: Vorrapan Chandee , Yoonbok Lee , Sheng-chi Liu , Maksym Radziwiłł

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 $\chi$ .

Keywords

dirichlet l-functions simple zeros generalized riemann hypothesis asymptotic large sieve pair correlation function analytic number theory

Cite

@article{arxiv.1211.6725,
  title  = {Simple zeros of primitive Dirichlet $L$-functions and the asymptotic large sieve},
  author = {Vorrapan Chandee and Yoonbok Lee and Sheng-chi Liu and Maksym Radziwiłł},
  journal= {arXiv preprint arXiv:1211.6725},
  year   = {2013}
}

Comments

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

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

Abstract

Keywords

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