Cesaro averages of Euler-like functions

TL;DR

This paper investigates the limiting behavior of Cesaro averages of Euler-like functions, which are multiplicative functions defined via prime factorization, providing new insights into their average properties.

Contribution

It introduces a method to compute the limit of Cesaro means for a class of Euler-like functions based on prime factorization.

Findings

Calculated the limit of Cesaro means for Euler-like functions

Provided explicit formulas for these averages

Enhanced understanding of multiplicative functions' average behavior

Abstract

By Euler-like function we mean a function defined on the positive integers and associating to $n$ the product, over all primes $p$ dividing $n$, of 1 plus (or minus) the inverse of $p$ to the power $s$. We calculate the limit of the Cesaro mean of these functions.