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 ${\mathbb{F}}_q[T]$ 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 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.

45 pages. Includes a new explicit characterization of decomposition and variance; e.g. Corollary 3.5

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Cited by in corpus (11)

• Higher moments of arithmetic functions in short intervals: a geometric perspective
• Mean values of arithmetic functions in short intervals and in arithmetic progressions in the large-degree limit
• The variance of the number of sums of two squares in ${\mathbb{F}}_q[T]$ in short intervals
• Chebotarev density theorem in short intervals for extensions of ${\mathbb{F}}_q(T)$
• A representation theory approach to integral moments of L-functions over function fields
• Variance of sums in arithmetic progressions of arithmetic functions associated with higher degree $L$-functions in ${\mathbb{F}}_q[t]$
• Variance of sums in arithmetic progressions of divisor functions associated with higher degree l-functions in ${\mathbb{F}}_q(t)$
• Correlation of arithmetic functions over ${\mathbb{F}}_q[T]$
• Magic squares, the symmetric group and Möbius randomness
• Variance of sums in short intervals and $L$-functions in ${\mathbb{F}}_q[t]$
• Prime polynomial values of quadratic functions in short intervals