Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2023; 60(1): 1-22

Online first article January 27, 2023      Printed January 31, 2023

https://doi.org/10.4134/BKMS.b210464

Copyright © The Korean Mathematical Society.

On Chowla's hypothesis implying that $L(s,\chi)>0$ for $s>0$ for real characters $\chi$

St\'ephane R. Louboutin

Aix Marseille Universit\'e, CNRS, Centrale Marseille, I2M.

Abstract

Let $L(s,\chi)$ be the Dirichlet $L$-series associated with an $f$-periodic complex function $\chi$. Let $P(X)\in {\mathbb C}[X]$. We give an expression for $\sum_{n=1}^f \chi (n)P(n)$ as a linear combination of the $L(-n,\chi)$'s for $0\leq n<\deg P(X)$. We deduce some consequences pertaining to the Chowla hypothesis implying that $L(s,\chi )>0$ for $s>0$ for real Dirichlet characters $\chi$. To date no extended numerical computation on this hypothesis is available. In fact by a result of R. C. Baker and H. L. Montgomery we know that it does not hold for almost all fundamental discriminants. Our present numerical computation shows that surprisingly it holds true for at least $65\%$ of the real, even and primitive Dirichlet characters of conductors less than $10^6$. We also show that a generalized Chowla hypothesis holds true for at least $72\%$ of the real, even and primitive Dirichlet characters of conductors less than $10^6$. Since checking this generalized Chowla's hypothesis is easy to program and relies only on exact computation with rational integers, we do think that it should be part of any numerical computation verifying that $L(s,\chi )>0$ for $s>0$ for real Dirichlet characters $\chi$. To date, this verification for real, even and primitive Dirichlet characters has been done only for conductors less than $2\cdot 10^5$.

Keywords: Real character, $L$-series, real zeros, Fekete polynomial

MSC numbers: Primary 11R18; Secondary 11R29, 11M06

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