Two problems related to the non-vanishing of \(L(1,\chi)\)
type: Item
instance of: scholarly article
title: Two problems related to the non-vanishing of \(L(1,\chi)\)
zbMATH Open document ID: 0929.11003
author: P. Codecà
author: Roberto Dvornicich
author: Umberto Zannier
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1999-03-07T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/93712
full work available at URL: jtnb10-1.html
review text: For a real number \(u\) let \(((u)):= u-[u]-1/2\) denote the centered fractional part of \(u\). The authors consider two different problems both leading to the study of the ranks of matrices with entries zero or \(((xy/q))\), where \(0\leq x\), \(y<q\). They observe also that these ranks are closely connected with the nonvanishing of the Dirichlet functions \(L(s,\chi)\) at \(s=1\). The first problem has its origin in the diophantine geometry and asks for the number of linearly independent functions of the form \[ f_{a,q}(n) =n/q+ \bigl((an/q) \bigr)- \biggl(\bigl(a+1) n/q\bigr) \biggr)=n/q+ g_{a, q}(n),\quad a,q\in\mathbb{Z},\;q>0. \] It is proved (Theorem 1) that the vector space generated by the functions \(g_{a,q}\) \((f_{a,q}\text{ resp.)}\) is equal to \([(q-3)/2]+d(q)\) \(([(q-1)/2]+d(q)\) resp.). The second problem arises from Fourier analysis and concerns functions of the form \[ g(x)=\sum^\infty_{n=1}{f(n)\over n}\cos 2 \pi nx, \] where \(f\) is complex-valued and periodic modulo \(q\) for a positive integer \(q\). It is proved that if all sums \(\delta_h(d):={1\over d}\sum^d_{a=1} h(a/d)\) vanish for \(d\geq 1\) then \(f\) is identically zero. Hence the sums \(\delta_h (d)\) uniquely determine \(h\) (Theorem 2).
Mathematics Subject Classification ID: 11A25
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 11M20
Mathematics Subject Classification ID: 42A16
zbMATH DE Number: 1229647
zbMATH Keywords: arithmetic functions
zbMATH Keywords: Dirichlet \(L\)-functions
zbMATH Keywords: Fourier series
zbMATH Keywords: ranks of matrices
reviewed by: J. Kaczorowski
MaRDI profile type: Publication
OpenAlex ID: W2066709666
cites work: Q4023361
cites work: A general commutative descent algebra
cites work: On an identity between infinite series of arithmetic functions
cites work: Q3943047
DOI: 10.5802/JTNB.218
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description: scientific article; zbMATH DE number 1229647