Toroidal families and averages of $L$-functions, II: cubic moments

TL;DR

This paper investigates the average of products of three special values of $L$-functions associated with Dirichlet characters, revealing connections to trace function estimates and solutions of monoidal equations over finite fields.

Contribution

It extends previous work on toroidal averages by analyzing cubic moments of $L$-functions and linking them to finite field combinatorics and trace function bounds.

Findings

Derived bounds for cubic moments of $L$-functions

Connected $L$-function averages to trace function estimates

Related solutions of monoidal equations to finite field problems

Abstract

Generalizing our previous work on ``toroidal averages'', we study the average of special values of $L$-functions of the form $L(1/2,\chi^a)L(1/2,\chi^b)L(1/2,\chi^c)$ for integers $a$, $b$ and $c$, where $\chi$ varies over Dirichlet characters of a given prime modulus. We highlight connections with estimates for bilinear forms of trace functions and with bounds for the number of solutions of monoidal equations in three variables in small boxes over finite fields.