A survey of the alternating sum-of-divisors function

TL;DR

This paper surveys the properties and open problems related to the alternating sum-of-divisors function, exploring its arithmetic and asymptotic behavior and extending it through multiplicativity.

Contribution

It provides a comprehensive overview of the alternating sum-of-divisors function's properties and highlights open problems in its study.

Findings

Detailed analysis of the function's arithmetic properties

Asymptotic behavior characterization

Identification of open problems in the field

Abstract

We survey arithmetic and asymptotic properties of the alternating sum-of-divisors function $\beta$ defined by $\beta(p^a)=p^a-p^{a-1}+p^{a-2}-...+(-1)^a$ for every prime power $p^a$ ($a\ge 1$), and extended by multiplicativity. Certain open problems are also stated.