Algebra & Number Theory

Arithmetic functions in short intervals and the symmetric group

Brad Rodgers

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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

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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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11M50: Relations with random matrices
Secondary: 11N37: Asymptotic results on arithmetic functions 11T55: Arithmetic theory of polynomial rings over finite fields

arithmetic in function fields random matrices the symmetric group


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.

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