Mixed moments of twisted $L$-functions

TL;DR

This paper derives an asymptotic formula with a power-saving error term for the mixed moments of twisted Dirichlet and automorphic L-functions across all admissible moduli, extending prior results to general moduli.

Contribution

It extends the asymptotic analysis of twisted L-function moments to all admissible moduli, achieving sharper error bounds comparable to prime modulus cases.

Findings

Established an asymptotic formula with a power-saving error term

Extended previous results to general moduli beyond prime moduli

Achieved error bounds as sharp as recent prime modulus bounds

Abstract

We establish an asymptotic formula with a power-saving error term for the twisted mixed moment of Dirichlet $L$-functions and automorphic $L$-functions twisted by all primitive characters modulo $q$, valid for all admissible moduli. As a special case, this extends the asymptotic result of Blomer, Fouvry, Kowalski, Michel, and Mili\'cevi\'c to general moduli, achieving an error term as sharp as the best bound recently proved by Khan and Zhang for prime moduli.