Arithmetic properties of the sum of divisors

TL;DR

This paper investigates the p-adic valuation of the divisor sum function, deriving formulas and bounds that relate to prime factorizations and special number classes like Mersenne primes.

Contribution

It provides explicit formulas for the p-adic valuation of the divisor function and establishes bounds with conditions for equality involving Mersenne primes and Diophantine equations.

Findings

Derived formulas for ν_p(σ(n)) involving prime divisors.

Established bounds for ν_2(σ(n)) and ν_p(σ(n)) with equality conditions.

Connected valuations to special prime structures and Diophantine solutions.

Abstract

The divisor function $\sigma(n)$ denotes the sum of the divisors of the positive integer $n$. For a prime $p$ and $m \in \mathbb{N}$, the $p$-adic valuation of $m$ is the highest power of $p$ which divides $m$. Formulas for $\nu_{p}(\sigma(n))$ are established. For $p=2$, these involve only the odd primes dividing $n$. These expressions are used to establish the bound $\nu_{2}(\sigma(n)) \leq \lceil\log_{2}(n) \rceil$, with equality if and only if $n$ is the product of distinct Mersenne primes, and for an odd prime $p$, the bound is $\nu_{p}(\sigma(n)) \leq \lceil \log_{p}(n) \rceil$, with equality related to solutions of the Ljunggren-Nagell diophantine equation.