Arithmetic functions in short intervals and the symmetric group

Abstract

We consider the variance of sums of arithmetic functions over random short intervals in the function field setting. Based on the analogy between factorizations of random elements of ${\mathbb{F}}_q\left[T\right]$ into primes and the factorizations of random permutations into cycles, we give a simple but general formula for these variances in the large $q$ limit for arithmetic functions that depend only upon factorization structure. From this we derive new estimates, quickly recover some that are already known, and make new conjectures in the setting of the integers.

In particular we make the combinatorial observation that any function of this sort can be explicitly decomposed into a sum of functions $u$ and $v$, depending on the size of the short interval, with $u$ making a negligible contribution to the variance, and $v$ asymptotically contributing diagonal terms only.

This variance evaluation is closely related to the appearance of random matrix statistics in the zeros of families of $L$-functions and sheds light on the arithmetic meaning of this phenomenon.