On certain arithmetic functions \(\tilde M(s; z_{1}; z_{2})\) associated with global fields: analytic properties
type: Item
instance of: scholarly article
title: On certain arithmetic functions \(\tilde M(s; z_{1}; z_{2})\) associated with global fields: analytic properties
zbMATH Open document ID: 1234.11119
DOI: 10.2977/PRIMS/36
author: Yasutaka Ihara
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2011-05-03T00:00:00Z
review text: In the paper, the analytic properties of the certain arithmetic functions \({\tilde M}(s;z_1,z_2)\) associated with global fields \(K\) are discussed. The author studies the general density functions \(M(s)| dx|\) on \(\mathbb R^d\), \(d=1,2,\dots\), with center 0, in particular, the best lower bound for the quantity \(\mu^{d/2}\nu\) (here \(\mu\) is the variance, \(\nu\) is the Plancherel volume). One of results is devoted to the investigation of the limit of \(\mu_\sigma\nu_\sigma\) and \(\mu_\sigma M_\sigma(\mu_\sigma^{1/2}w)\) as \(\sigma \to 1/2, +\infty\). The analytic continuation to the space \(\{\text{Re}\,s >0\}\setminus\{\rho/(2n); n \in \mathbb N, \zeta(\rho=0 \;\text{or} \;\infty)\}\times\mathbb C^2\) of the function \({\tilde M}(s;z_1,z_2)\) is proved (here \(\zeta(s)\) is the zeta-function of \(K\) without factors of finite set of prime divisors of \(K\) including all archimedean primes in the number field case). The last part of the paper is devoted to the studies of the rapid decay property of \(|{\tilde M}_\sigma(z)|^2\) with attention when \(\sigma\) is arbitrarily close to \(1/2\) and \(| z|\) is unbounded.
reviewed by: Roma Kačinskaitė
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11R42
Mathematics Subject Classification ID: 14H25
Mathematics Subject Classification ID: 60E10
Mathematics Subject Classification ID: 60E15
zbMATH DE Number: 5883096
zbMATH Keywords: \(L\)-function
zbMATH Keywords: arithmetic function
zbMATH Keywords: critical point
zbMATH Keywords: Plancherel equality
zbMATH Keywords: probabilistic inequality
MaRDI profile type: Publication
cites work: Q3393096
cites work: On ``\(M\)-functions'' closely related to the distribution of \(L'/L\)-values
cites work: Q3081687
cites work: Q3582055
cites work: Distribution Functions and the Riemann Zeta Function
cites work: Q3290778
description: scientific article; zbMATH DE number 5883096