Some remarks on sum of Euler's totient function

TL;DR

This paper investigates the sum of Euler's totient function over integers up to n, focusing on cases where a prime p divides n, providing insights into the function's behavior in these scenarios.

Contribution

It offers a new result concerning the sum of Euler's totient function specifically when a prime divides n, extending existing understanding of the function's properties.

Findings

Derived a formula for the sum of totient function when p divides n

Enhanced understanding of totient sum behavior in prime-divisible cases

Provides theoretical insights into totient function sums

Abstract

Euler's totient function counts the positive integers up to a given integer n that are relatively prime to n. The aim of this article is to give a result about the sum of euler's totient function from k equal 1 to n whene p divides n and p a prime number.