An analytic approach to the remainder terms in the asymptotic formulas -- the Volterra integral equation, the Whittaker function

TL;DR

This paper develops an analytical method to study the remainder terms in asymptotic formulas related to arithmetic functions, employing Volterra integral equations and special functions like the Whittaker function.

Contribution

It introduces a novel approach using Volterra integral equations and special functions to analyze remainder terms in asymptotic formulas for arithmetic functions.

Findings

Solution of the Volterra integral equation for the remainder term.

Application to the Euler totient function's asymptotic formula.

Analytic properties of functions over zeros of generalized L-functions.

Abstract

In the present paper, firstly, we consider the Volterra integral equation of second type for a remainder term in an asymptotic formula of an arithmetic function which satisfies some special conditions and obtained a solution of the equation. The method using there is applied to the remainder term in an asymptotic formula of the associated Euler totient function. Secondly, we consider a function defined a series over all non-trivial zeros of the generalized $L$-functions. We proved some analytic properties which satisfy some conditions. In particular, we use the Whittaker function which is kind of the confluent hypergeometric function there.