The fourth power mean of Dirichlet L-functions in $${\mathbb {F}}_q [T]$$

Abstract

AbstractWe 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 $${\mathbb {F}}_{q}[T]$$ F q [ T ] . We consider the behaviour as $${{\,\mathrm{deg}\,}}R \rightarrow \infty $$ deg R → ∞ 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.