Variance of sums in arithmetic progressions of divisor functions associated with higher degree L-functions in 𝔽q[t

Abstract

We compute the variances of sums in arithmetic progressions of generalized [Formula: see text]-divisor functions related to certain [Formula: see text]-functions in [Formula: see text], in the limit as [Formula: see text]. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when [Formula: see text], in terms of matrix integrals, which may be evaluated. Our results extend those obtained previously in the special case corresponding to the usual [Formula: see text]-divisor function, when the [Formula: see text]-function in question has degree one. They illustrate the role played by the degree of the [Formula: see text]-functions; in particular, we find qualitatively new behavior when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over [Formula: see text], and we illustrate them by examining in some detail the generalized [Formula: see text]-divisor functions associated with the Legendre curve.