Alternating sums concerning multiplicative arithmetic functions

TL;DR

This paper derives asymptotic formulas for alternating sums involving classical multiplicative functions and their variants, improving error estimates and proposing open problems in the field.

Contribution

It provides new asymptotic formulas for alternating sums of multiplicative functions and their variants, with improved error terms and insights into related open problems.

Findings

Asymptotic formulas for sums involving Euler's totient, Dedekind, divisor, and gcd-sum functions.

Improved error estimates over previous results by Bordellès and Cloitre.

Formulation of open problems related to these sums.

Abstract

We deduce asymptotic formulas for the alternating sums $\sum_{n\le x} (-1)^{n-1} f(n)$ and $\sum_{n\le x} (-1)^{n-1} \frac1{f(n)}$, where $f$ is one of the following classical multiplicative arithmetic functions: Euler's totient function, the Dedekind function, the sum-of-divisors function, the divisor function, the gcd-sum function. We also consider analogs of these functions, which are associated to unitary and exponential divisors, and other special functions. Some of our results improve the error terms obtained by Bordell\`{e}s and Cloitre. We formulate certain open problems.