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Equivalence between the Functional Equation and Voronoï-type summation identities for a class of $L$-Functions

Arindam Roy, Jagannath Sahoo, Akshaa Vatwani

Abstract

To date, the best methods for 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-type summation 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.

Equivalence between the Functional Equation and Voronoï-type summation identities for a class of $L$-Functions

Abstract

To date, the best methods for 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 can be obtained with ``nice" test functions , provided is an ``arithmetic function". These arithmetic functions are called so because they are expected to appear as coefficients of some -functions satisfying certain properties. It has been well-known that the functional equation for a general -function can be used to derive a Voronoï-type summation identity for that -function. In this article, we show that such a Voronoï-type summation identity in fact endows the -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.

Paper Structure

This paper contains 8 sections, 16 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Prerequisites
  3. Proof of Theorem \ref{['thm:FE implies mod rel1']}
  4. Proof of Theorem \ref{['thm:mod rel1 implies FE']}
  5. Proof of Theorem \ref{['thm:FE implies Rieszsum']}
  6. Proof of Theorem \ref{['thm:riesz sum implies modreln 2']}
  7. Proof of Theorem \ref{['thm:modreln2 implies FE']}
  8. Special cases

Key Result

Theorem 1.1

Let $\phi(s)$ and $\psi(s)$ be two Dirichlet series in the class $\mathcal{C}$. Then the functional equation where $F(s)= \phi(s)\prod_{i=1}^r \Gamma\left(\alpha_i s+\beta_i \right)$, $G(s) = \psi(s)\prod_{i=1}^r \Gamma\left(\alpha_i s+\beta_i \right)$, implies the modular relation where for $x,a>0$, $P(x)$ is a residual function given by and $C$ denotes a circle of finite radius, lying inside

Theorems & Definitions (25)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • remark 1
  • remark 2
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Definition 2.1
  • ...and 15 more