Variance of sums in arithmetic progressions of divisor functions associated with higher degree l-functions in ${\mathbb{F}}_q(t)$

Abstract

We compute the variances of sums in arithmetic progressions of generalised k-divisor functions related to certain L-functions in ${\mathbb{F}}_q(t)$, in the limit as $q\to \infty$. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when $q\to \infty$, in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual $k$-divisor function, when the L-function in question has degree one. They illustrate the role played by the degree of the L-functions; in particular, we find qualitatively new behaviour when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over ${\mathbb{F}}_q(t)$, and we illustrate them by examining in some detail the generalised $k$-divisor functions associated with the Legendre curve.