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Equivalence between the functional equation and Voronoï-type summation identities for a class of L-functions

Part of: Zeta and $L$-functions: analytic theory

Published online by Cambridge University Press:  25 November 2024

Arindam Roy Open the ORCID record for Arindam Roy [Opens in a new window] ,
Jagannath Sahoo  and
Akshaa Vatwani Open the ORCID record for Akshaa Vatwani [Opens in a new window]
Arindam Roy
Affiliation:
Department of Mathematics, University of North Carolina at Charlotte, Charlotte, 28223 NC, USA (arindam.roy@charlotte.edu)
Jagannath Sahoo
Affiliation:
Department of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, 382355 Gujarat, India (jagannath.sahoo@iitgn.ac.in, akshaa.vatwani@iitgn.ac.in) (corresponding author)
Akshaa Vatwani
Affiliation:
Department of Mathematics, Indian Institute of Technology Gandhinagar, Palaj, Gandhinagar, 382355 Gujarat, India (jagannath.sahoo@iitgn.ac.in, akshaa.vatwani@iitgn.ac.in) (corresponding author)
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Abstract

To date, the bestmethodsfor estimating the growth of mean values of arithmetic functions rely on the Voronoï summation formula. By noticing a general pattern in the proof of his summation formula, Voronoï postulated that analogous summation formulas for $\sum a(n)f(n)$ can be obtained with ‘nice’ test functions f(n), provided a(n) is an ‘arithmetic function’. These arithmetic functions a(n) are called so because they are expected to appear as coefficients of some L-functions satisfying certain properties. It has been well-known that the functional equation for a general L-function can be used to derive a Voronoï-type summation identity for that L-function. In this article, we show that such a Voronoï-typesummation identity in fact endows the L-function with some structural properties, yielding in particular the functional equation. We do this by considering Dirichlet series satisfying functional equations involving multiple Gamma factors and show that a given arithmetic function appears as a coefficient of such a Dirichlet series if and only if it satisfies the aforementioned summation formulas.

Keywords

Hecke functional equationmodular relationsVoronoï summation formulaRiesz-sum identititesL-functions

MSC classification

Primary: 11M06: $zeta (s)$ and $L(s, chi)$
Secondary: 11M26: Nonreal zeros of $zeta (s)$ and $L(s, chi)$; Riemann and other hypotheses 11M41: Other Dirichlet series and zeta functions

Information

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 41
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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