Variance of Squarefull Numbers in Short Intervals

TL;DR

This paper investigates the distribution variance of squarefull numbers within short intervals, establishing that almost all such intervals contain a predictable number of these numbers based on a specific asymptotic formula.

Contribution

It provides a new variance analysis of squarefull numbers in short intervals and proves almost sure distribution results for these numbers.

Findings

Almost all short intervals contain about rac{/2}{2 } x^ heta squarefull numbers.

The number of squarefull numbers in short intervals follows a specific asymptotic distribution.

The variance of the count of squarefull numbers in short intervals is characterized.

Abstract

In this paper, we study the variance of the number of squarefull numbers in short intervals. As a result, we are able to prove that, for any $0 < \theta < 1/2$, almost all short intervals $(x, x + x^{1/2 + \theta}]$ contain about $\frac{\zeta(3/2)}{2 \zeta(3)} x^\theta$ squarefull numbers.