Equivalence between the Functional Equation and Vorono\"ı-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 Vorono\"ı summation formula. By noticing a general pattern in the proof of his summation formula, Vorono\"ı 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 Vorono\"ı-type summation identity for that $L$-function. In this article, we show that such a Vorono\"ı-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.

32 pages, Published in Proceedings of the Royal Society of Edinburgh: Section A, Mathematics