A note on the $q$-adic valuation of $σ_k(n)$

Abstract

In this note, we obtain an exact formula for the $q$-adic valuation of $σ_k(n)$ where $q$ is an odd prime, allowing us to derive an explicit upper bound which is asymptotically better than the previous bound obtained by Zhao when $n$ is large and $k \geqslant q-2$. The key parts are played by the LTE lemma and the use of cyclotomic polynomials.