p -ADIC VALUATION OF $\sigma _k(n)$

Abstract For any integer k and any positive integer n , let $\sigma _k(n)=\sum _{d\mid n}d^k$ . For any prime p and any positive integer m , let $\nu _p(m)$ be the largest integer $\alpha $ such that $p^\alpha \mid m$ and let $\lceil x\rceil $ denote the least integer not less than x . In 2021, Amdeberhan et al. [‘Arithmetic properties of the sum of divisors’, J. Number Theory 223 (2021), 325–349] proved that $\nu _2(\sigma _1(n))\le \lceil \log _2n\rceil $ for any positive integer n and that $\nu _p(\sigma _1(n))\le \lceil \log _pn\rceil $ for any odd prime p if n satisfies some conditions. Recently, Zhao and Chen [‘ p -adic valuation of the sum of divisors’, Front. Math. 20 (4) (2025), 795–827] proved this unconditionally. We generalise these results to all k : for any prime p , any n and any $k\ge 2$ , $\nu _p(\sigma _k(n))\le \lceil k\log _p n\rceil .$ Let $p^\star $ be an odd prime. We also prove that there are an integer $k\ge 2$ and a prime q satisfying $\nu _{p^\star }(\sigma _k(q))=\lceil k\log _{p^\star }q\rceil $ if and only if $p^\star $ is a Fermat prime.