Algebra & Number Theory
- Algebra Number Theory
- Volume 12, Number 5 (2018), 1243-1279.
Arithmetic functions in short intervals and the symmetric group
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 into primes and the factorizations of random permutations into cycles, we give a simple but general formula for these variances in the large 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 and , depending on the size of the short interval, with making a negligible contribution to the variance, and asymptotically contributing diagonal terms only.
This variance evaluation is closely related to the appearance of random matrix statistics in the zeros of families of -functions and sheds light on the arithmetic meaning of this phenomenon.
Algebra Number Theory, Volume 12, Number 5 (2018), 1243-1279.
Received: 30 September 2017
Revised: 29 January 2018
Accepted: 18 March 2018
First available in Project Euclid: 14 August 2018
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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