On two conjectures of Sierpiński concerning the arithmetic functions $\sigma$ and $\varphi$

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 Sierpiński from 1955. Additionally, it is shown that for every positive even $k$, there is a number $m$ for which the equation $\varphi (x)=m$ has exactly $k$ solutions, where $\varphi$ is Euler's function, making progress toward another conjecture of Sierpiński from 1955.

published in 1999