On the geometric mean of the values of positive multiplicative arithmetical functions

TL;DR

This paper derives an asymptotic expansion for the geometric mean of positive strongly multiplicative functions, extending understanding of their average behavior under specific growth conditions.

Contribution

It provides a new asymptotic formula for the geometric mean of positive strongly multiplicative functions with specified prime behavior.

Findings

Derived asymptotic expansion for the geometric mean

Extended results to functions with specific prime growth conditions

Enhanced understanding of multiplicative function averages

Abstract

In this paper we obtain asymptotic expansion for the geometric mean of the values of positive strongly multiplicative function $f$ satisfying $f(p)=\alpha(d)\,p^d+O(p^{d-\delta})$ for any prime $p$ with $d$ real and $\alpha(d),\delta>0$.