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

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.