On the $2$-adic valuation of ${\sigma }_k(n)$

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 $n\ge 2$, $${\nu }_2({\sigma }_k(n))\le \{\begin{array}{ll}\lceil {\log }_{2}n\rceil ,& \text{if }k\text{ is odd},\\ \lfloor {\log }_{2}n\rfloor ,& \text{if }k\text{ is even}\mathrm{.}\end{array}$$ 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 $n$.