General divisor function inequalities and the third cumulant

TL;DR

This paper extends divisor sum inequalities to general cases, introduces entropy bounds for geometric distributions, and analyzes probabilities related to the third cumulant of random variables.

Contribution

It generalizes a divisor sum inequality, establishes new entropy bounds, and links third cumulant conditions to probability estimates.

Findings

Extended divisor sum lower bounds to non-squarefree integers

Proved a lower bound on the entropy of finite support geometric distributions

Derived probability bounds for variables less than their mean based on third cumulant conditions

Abstract

We extend a lower bound of Munshi on sums over divisors of a number $n$ which are less than a fixed power of $n$ from the squarefree case to the general case. In the process we prove a lower bound on the entropy of a geometric distribution with finite support, as well as a lower bound on the probability that a random variable is less than its mean given that it satisfies a natural condition related to its third cumulant.