On the variance of sums of divisor functions in short intervals

TL;DR

This paper investigates the variance of sums of the k-fold divisor function within short intervals, deriving asymptotic formulas for specific cases under the Lindelöf hypothesis, advancing understanding of divisor function distribution.

Contribution

It provides new asymptotic formulas for the variance of divisor sums in short intervals for k=3 and k≥4, assuming the Lindelöf hypothesis, extending previous results.

Findings

Asymptotic formulas established for k=3 and k≥4

Results depend on the Lindelöf hypothesis

Enhanced understanding of divisor function distribution in short intervals

Abstract

Given a positive integer $n$ the $k$-fold divisor function $d_k(n)$ equals the number of ordered $k$-tuples of positive integers whose product equals $n$. In this article we study the variance of sums of $d_k(n)$ in short intervals and establish asymptotic formulas for the variance of sums of $d_k(n)$ in short intervals of certain lengths for $k=3$ and for $k \ge 4$ under the assumption of the Lindel\"of hypothesis.