Approximate Functional Equation for the Product of Functions and Divisor Problem

TL;DR

The paper addresses deriving an $\text{approximate functional equation}$ for the product of functions defined by Dirichlet or generalized Dirichlet series in certain half-planes, with an explicit remainder term, in the context of the divisor problem. It introduces a simple elementary approach that relies on a highly generalized identity modeled after Motohashi's framework. The main contribution is a concise derivation of the $\text{approximate functional equation}$ for the product along with an explicit remainder term, enabling sharper analysis of divisor-type phenomena. The result provides a practical analytic tool for studying products of Dirichlet-series-type objects in analytic number theory.

Abstract

For functions defined via Dirichlet/generalized Dirichlet series in some half planes of the complex plane, we give a new simple elementary approach to obtain an approximate functional equation for the product of functions with explicit remainder term. We do this using a highly generalized identity following Motohashi's approach.