On the $2$-adic valuation of $\sigma_k(n)$

Abstract

For a positive integer $k$, let \[ \sigma_k(n)=\sum_{d\mid n} d^k \] be the divisor function of order $k$, and let $\nu_p(m)$ denote the $p$-adic valuation of an integer $m$. Motivated by recent work on the $p$-adic valuation of $\sigma_k(n)$, we study $\nu_2(\sigma_k(n))$ in detail. We prove that, for every integer $n\ge 2$, \[ \nu_2(\sigma_k(n)) \le \begin{cases} \lceil \log_2 n \rceil, & \text{if $k$ is odd},\\[1mm] \lfloor \log_2 n \rfloor, & \text{if $k$ is even}. \end{cases} \] These bounds are best possible. More precisely, if $k$ is odd, then equality holds if and only if $n$ is a product of distinct Mersenne primes; if $k$ is even, then equality holds if and only if $n=3$. We also obtain an explicit formula for $\nu_2(\sigma_k(n))$ in terms of the prime factorization of $n$.