On the variance of sums of divisor functions in short intervals

Lester, Stephen

doi:10.1090/proc/12914

On the variance of sums of divisor functions in short intervals
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Proc. Amer. Math. Soc. 144 (2016), 5015-5027 Request permission

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 Lindelöf hypothesis.

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