On the Volterra integral equation for the remainder term in the asymptotic formula on the associated Euler totient function

TL;DR

This paper studies a Volterra integral equation related to the error term in the asymptotic formula for the Euler totient function, providing solutions and splitting the error into arithmetic and analytic parts.

Contribution

It introduces a novel approach to analyze the remainder term in the Euler totient asymptotics via Volterra integral equations and decomposes the error into two distinct components.

Findings

Solved the Volterra integral equation for the remainder term.

Decomposed the error into arithmetic and analytic parts.

Provided insights into the structure of the error in Euler totient asymptotics.

Abstract

This paper, first, we consider the Volterra integral equation for the remainder term in the asymptotic formula for the associated Euler totient function. Secondly, we solve the Volterra integral equation and we split the error term in the asymptotic formula for the associated Euler totient function into two summands called arithmetic and analytic part respectively.