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.
Keywords
arithmetic in function fields, random matrices, the
symmetric group
