The fourth moment of derivatives of Dirichlet L-functions in function fields

Abstract

We obtain the asymptotic main term of moments of arbitrary derivatives of L-functions in the function field setting. Specifically, we obtain the first, second, and mixed fourth moments. The average is taken over all non-trivial characters of a prime modulus $$Q \in {\mathbb {F}}_q [T]$$ Q ∈ F q [ T ] , and the asymptotic limit is as $${{\,\mathrm{deg}\,}}Q \longrightarrow \infty $$ deg Q ⟶ ∞ . This extends the work of Tamam who obtained the asymptotic main term of low moments of L-functions, without derivatives, in the function field setting. It is also the function field q-analogue of the work of Conrey, who obtained the fourth moment of derivatives of the Riemann zeta-function.

Moments of Dirichlet $L$-functions to a fixed modulus over function fields

“…Proof. The expression in (2.12) can be found on [30, p. 189] and the expression in (2.13) follows by combining Lemmas 3.10 and 3.11 in [2].…”

“…The result for the fourth moment is extended by J. C. Andrade and M. Yiasemides [3] to hold for a general polynomial Q. In [2], Andrade and Yiasemides further studied mixed fourth moments of all derivatives of the L-functions under consideration at the central point.…”