On summatory arithmetic functions and a Volterra Integral equation

TL;DR

This paper derives asymptotic behaviors of classical summatory arithmetic functions, introduces a novel Volterra integral equation linked to these functions, and explores their applications in number theory.

Contribution

It presents a new Volterra integral equation solved by summatory functions and connects it to existing number theoretic formulas, offering fresh analytical tools.

Findings

Asymptotic formulas for $\psi(x)$ and related functions

A new Volterra integral equation linked to summatory functions

Applications to number theoretic formulas

Abstract

We obtain asymptotic results for well known summatory arithmetic functions, such as $\psi(x),$ and establish connections to new summatory functions. A new Volterra integral equation is offered, which is solved by summatory arithmetic functions. We conclude with some further integral formulas and provide number theoretic formulas as applications.