On the solution of the Volterra integral equation of second type for the error term in an asymptotic formula for arithmetical functions

TL;DR

This paper investigates solving the Volterra integral equation of second type related to the error term in asymptotic formulas for arithmetical functions, providing conditions under which solutions can be explicitly obtained.

Contribution

It introduces a method to solve the Volterra integral equation for the error term in asymptotic formulas of arithmetical functions under specific conditions on the functions involved.

Findings

Solution can be explicitly obtained under certain conditions on a(n)

Provides a new approach to analyze error terms in asymptotic formulas

Enhances understanding of integral equations in number theory

Abstract

This paper, we first consider the pair of complex-valued arithmetical functions (a(n),b(n)) satisfying. We prove that the solution of the Volterra integral equation of second type for the error term in the asymptotic formula for b(n) can be obtained when a(n) satisfies some special condition.