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

TL;DR

This paper derives an exact formula for the q-adic valuation of the sum-of-divisors function, providing improved bounds for large n and k, using LTE lemma and cyclotomic polynomials.

Contribution

It introduces a precise formula for the q-adic valuation of _k(n) and improves existing bounds for large n and k.

Findings

Exact formula for q-adic valuation of _k(n)

Explicit upper bound better than previous results for large n

Utilizes LTE lemma and cyclotomic polynomials effectively

Abstract

In this note, we obtain an exact formula for the $q$-adic valuation of $\sigma_k(n)$ 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.