Evaluation of the Li coefficients on function fields and applications
type: Item
instance of: scholarly article
title: Evaluation of the Li coefficients on function fields and applications
zbMATH Open document ID: 1429.11174
author: Kamel Mazhouda
author: Lejla Smajlovic
published in: European Journal of Mathematics
publication date: 2019-06-26T00:00:00Z
review text: The classical Riemann hypothesis is equivalent to several statements. In particular, it is equivalent to $\lambda_n\geq 0$ for $n\geq 1,$ where $\lambda_n:= \sum_{\rho}^*\big[1-\big(1-\frac{1}{\rho}\big)^n\big]$. The sum runs through the nontrivial zeros of the Riemann zeta function and $^*$ denotes that the sum is taken in the sense of the limit $\lim_{|(\rho)|\to\infty}$. The Riemann hypothesis for global function fields was proved by A. Weil in 1940--1941. The $n$-th Li coefficient for a global function field $K$ is defined as \[ \lambda_K(n)= \mathop{{\sum}^*}_{\rho \in {\mathcal Z}(K)} \big(1-\big(1-\tfrac{1}{\rho}\big)^n\big)=\lim_{T\to \infty}\sum_{\substack{ {\rho\in{\mathcal Z}(K)}\\ |\mathrm{Im}(\rho)|\leq T}} \big(1-\big(1-\tfrac{1}{\rho}\big)^n\big)\] where ${\mathcal Z}(K)$ is the set of zeros of the Riemann zeta function $\zeta_K(s)$. We have that the Riemann hypothesis is equivalent to $\lambda_K(n)\geq 0$ for $n\geq 1$. In the paper under review, the authors first derive several formulas to compute $\lambda_K(n)$ (Theorems 3.2, 4.1, 4.2 and Corollary 3.3). Second, they apply the above formulas and the inequalities $\lambda_K(n)\geq 0$ to approximate some arithmetic invariants of the field $K$ (Corollaries 5.1, 5.2, 5.3 and Example 5.4). For instance, they show that \[ g h_K\max\big\{1,2\big(1-\frac{2}{\log q}\big) \big\}\leq L'(1)\leq 2 gh_K\big(1+\frac{2}{\log q}\big)\] where $L$ is the $L$--function associated to $K$, $g$ is the genus and $h_K$ is the class number of $K$.
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11R58
zbMATH DE Number: 7073696
zbMATH Keywords: global function fields
zbMATH Keywords: Riemann hypothesis
zbMATH Keywords: approximation of $L$-functions
reviewed by: Gabriel D. Villa Salvador
MaRDI profile type: Publication
full work available at URL: s40879-018-0212-6
OpenAlex ID: W2786965142
cites work: Complements to Li's criterion for the Riemann Hypothesis
cites work: On a Li-type criterion for zero-free regions of certain Dirichlet series with real coefficients
cites work: On the modified Li criterion for a certain class of \(L\)-functions
cites work: Li coefficients for automorphic \(L\)-functions
cites work: The positivity of a sequence of numbers and the Riemann hypothesis
cites work: On the Li coefficients for the Dirichlet \(L\)-functions
cites work: On relations equivalent to the generalized Riemann hypothesis for the Selberg class
cites work: Q3998729
cites work: On Li's coefficients for the Rankin-Selberg \(L\)-functions
cites work: On asymptotic behavior of generalized Li coefficients in the Selberg class
cites work: Corrigendum and addendum to: Li's criterion and the Riemann hypothesis for the Selberg class
cites work: The Li criterion and the Riemann hypothesis for the Selberg class. II
cites work: On Li's criterion for the Riemann hypothesis for the Selberg class
cites work: Q5490909
cites work: Sharpenings of Li's criterion for the Riemann hypothesis
cites work: Q5794285
DOI: 10.1007/S40879-018-0212-6
description: scientific article; zbMATH DE number 7073696