On a problem of Pongsriiam on the sum of divisors, II

TL;DR

This paper investigates the distribution of positive integers where the sum of divisors function satisfies a specific inequality, providing bounds on their density and addressing a problem posed by Pongsriiam.

Contribution

It proves that the set of integers with $\sigma(30n+1)\geq\sigma(30n)$ has a density less than 0.0371813, partially answering Pongsriiam's recent problem.

Findings

Density of such integers is less than 0.0371813.

Provides partial resolution to Pongsriiam's problem.

Analyzes the sum of divisors function in a specific arithmetic context.

Abstract

For any positive integer $n$, let $\sigma (n)$ be the sum of all positive divisors of $n.$ In this paper, it is proved that the set of positive integers $ n $ for which $ \sigma(30n+1)\geq \sigma(30n) $ has a density less than $ 0.0371813, $ which answers a recent problem of Pongsriiam in part.