## Algebra & Number Theory

### 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 $F q [ T ]$ 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.

#### Article information

Source
Algebra Number Theory, Volume 12, Number 5 (2018), 1243-1279.

Dates
Revised: 29 January 2018
Accepted: 18 March 2018
First available in Project Euclid: 14 August 2018

https://projecteuclid.org/euclid.ant/1534212102

Digital Object Identifier
doi:10.2140/ant.2018.12.1243

Mathematical Reviews number (MathSciNet)
MR3840876

Zentralblatt MATH identifier
06921175

#### Citation

Rodgers, Brad. Arithmetic functions in short intervals and the symmetric group. Algebra Number Theory 12 (2018), no. 5, 1243--1279. doi:10.2140/ant.2018.12.1243. https://projecteuclid.org/euclid.ant/1534212102

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