On two conjectures of Sierpi\'nski concerning the arithmetic functions $\sigma$ and $\phi$

TL;DR

This paper proves that for any positive integer k, there exists a number m such that the divisor sum function σ(x)=m has exactly k solutions, and for every positive even k, a similar result holds for Euler's totient function φ(x), advancing Sierpiński's conjectures.

Contribution

It confirms Sierpiński's conjectures by showing the existence of numbers with exactly k solutions for σ and φ functions, for all positive integers and even k respectively.

Findings

For any positive integer k, there exists m with exactly k solutions to σ(x)=m.

For every positive even k, there exists m with exactly k solutions to φ(x)=m.

Progress towards Sierpiński's conjectures on arithmetic functions.

Abstract

Let $\sigma(n)$ denote the sum of the positive divisors of $n$. We prove that for any positive integer $k$, there is a number $m$ for which the equation $\sigma(x)=m$ has exactly $k$ solutions, settling a conjecture of Sierpi\'nski from 1955. Additionally, it is shown that for every positive even $k$, there is a number $m$ for which the equation $\phi(x)=m$ has exactly $k$ solutions, where $\phi$ is Euler's function, making progress toward another conjecture of Sierpi\'nski from 1955.