On the Tong-type identity and the mean square of the error term for an extended Selberg class

TL;DR

This paper generalizes Tong's method for the mean square of error terms to a broader class of Dirichlet series satisfying functional equations, extending results within the Selberg class and providing an analogue to Voronoi's formula.

Contribution

It introduces a Tong-type identity applicable to functions in the Selberg class, enabling asymptotic analysis of error terms for these functions.

Findings

Established asymptotic formulas for mean square error terms in the Selberg class.

Generalized Tong's method to a wider class of Dirichlet series.

Provided an analogue of Voronoi's formula for these functions.

Abstract

In 1956, Tong established an asymptotic formula for the mean square of the error term in the summatory function of the Piltz divisor function $d_3(n).$ The aim of this paper is to generalize Tong's method to a class of Dirichlet series that satisfy a functional equation. As an application, we can establish the asymptotic formulas for the mean square of the error terms for a class of functions in the well-known Selberg class. The Tong-type identity and formula established in this paper can be viewed as an analogue of the well-known Vorono\"i's formula.