| Summary: | We compute the variances of sums in arithmetic progressions of generalised 𝑘-divisor functions related to certain 𝐿-functions in 𝔽q[𝑡], in the limit as q → ∞. This is achieved by making use of recently established equidistribution results for the associated Frobenius conjugacy classes. The variances are thus expressed, when q → ∞, 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 𝔽q[𝑡], and we illustrate them by examining in some detail the generalised 𝑘-divisor functions associated with the Legendre curve.
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