Variance Of Sums In Arithmetic Progressions Of Divisor Functions Associated With Higher Degree L-Functions In F-Q[T]

We compute the variances of sums in arithmetic progressions of generalized k-divisor functions related to certain L-functions in F-q[t], in the limit as q -> infinity. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when q -> infinity, 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 behavior when the degree exceeds one. Our calculations apply, for example, to elliptic curves defined over F-q[t], and we illustrate them by examining in some detail the generalized k-divisor functions associated with the Legendre curve.