On the distribution of $v_p(\sigma(n))$

Abstract: For a positive integer $m$ and a prime $p$, we write $\sigma(m) := \sum_{d \mid m} d$ for the sum of the divisors of $m$, and $v_p(m) := \max\{ k \in \mathbf{Z}_{\ge 0} : p^k \mid m \}$ for the $p$-adic valuation of $m$, i.e., the exponent of $p$ in the prime factorization of $m$. For each prime $p$, we give an asymptotic expression for the count $$ \#\{ n \le x : v_p(\sigma(n)) = k \} $$ as $x\to\infty$, uniformly for $k \ll \log\log{x}$. We then deduce an asymptotic for the count of $n \le x$ such that $v_p(\sigma(n)) < v_p(n)$ as $x \to \infty$. \\ This talk is based on ongoing work with Paul Pollack.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)