Vol. 16, No. 5, 2022

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The mean values of cubic $L$-functions over function fields

Chantal David, Alexandra Florea and Matilde Lalín

Vol. 16 (2022), No. 5, 1259–1326

We obtain an asymptotic formula for the mean value of L-functions associated to cubic characters over 𝔽q[T]. We solve this problem in the non-Kummer setting when q 2(mod3) and in the Kummer setting when q 1(mod3). In the Kummer setting, the mean value over the complete family of cubic characters was never addressed in the literature (over number fields or function fields). The proofs rely on obtaining precise asymptotics for averages of cubic Gauss sums over function fields, which can be studied using the pioneer work of Kubota. In the non-Kummer setting, we display some explicit (and unexpected) cancellation between the main term and the dual term coming from the approximate functional equation of the L-functions. Exhibiting the cancellation involves evaluating sums of residues of a variant of the generating series of cubic Gauss sums.

moments over function fields, cubic twists, nonvanishing
Mathematical Subject Classification
Primary: 11M06, 11M38, 11R16
Secondary: 11R58
Received: 19 May 2021
Accepted: 26 July 2021
Published: 16 August 2022
Chantal David
Department of Mathematics and Statistics
Concordia University
Montreal, QC
Alexandra Florea
Department of Mathematics
University of California
Irvine, CA
United States
Matilde Lalín
Département de mathématiques et de statistique
Université de Montréal
Montreal, QC