Abstract
We prove results on moments of L-functions in the function field setting, where the moment averages are taken over primitive characters of modulus R, where R is a polynomial in {mathbb {F}}_{q}[T]. We consider the behaviour as {{,mathrm{deg},}}R rightarrow infty and the cardinality of the finite field is fixed. Specifically, we obtain an exact formula for the second moment provided that R is square-full, an asymptotic formula for the second moment for any R, and an asymptotic formula for the fourth moment for any R. The fourth moment result is a function field analogue of Soundararajan’s result in the number field setting that improved upon a previous result by Heath-Brown. Both the second and fourth moment results extend work done by Tamam in the function field setting who focused on the case where R is prime. As a prerequisite for the fourth moment result, we obtain, for the special case of the divisor function, the function field analogue of Shiu’s generalised Brun–Titchmarsh theorem.
Highlights
The study of moments of families L-functions is a central theme in analytic number theory
These moments are connected to the famous Lindelöf hypothesis for such L-functions and have many applications in analytic number theory
In this paper we prove the function field analogue of Soundararajan’s fourth moment result, which is an extension of Tamam’s fourth moment result
Summary
The study of moments of families L-functions is a central theme in analytic number theory. Young [11] obtained explicit lower order terms for the case where q is an odd prime and was able to establish the full polynomial expansion for the fourth moment of the associated Dirichlet L-functions. We obtain an asymptotic main term for the second moment This generalises Tamam’s result in that her result is for all primitive characters of prime modulus, whereas our result is for primitive characters of any modulus. By considering only square-full moduli, we obtain an exact formula
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