The fourth moment of Dirichlet $L$-functions at the central value

TL;DR

This paper proves the predicted asymptotic formula for the fourth moment of Dirichlet L-functions at the central point for general moduli, extending previous prime modulus results using divisor sums and Kloosterman sum bounds.

Contribution

It establishes the asymptotic formula for the fourth moment of Dirichlet L-functions at the central value for all moduli, not just prime, using novel divisor sum analysis and bounds on Kloosterman sums.

Findings

Confirmed the conjectured asymptotic formula for general moduli

Developed bounds for double sums in Kloosterman sums

Analyzed the $ ext{D}_q$-function to extract main terms

Abstract

The asymptotic formula of the fourth moment of Dirichlet $L$-functions at the central value was predicted in a conjecture by J. B. Conrey, D. W. Farmer, J. P. Keating, M. O. Rubinstein, and N. C. Snaith, and the prime moduli case was proved by M. P. Young in 2011. This work establishes this asymptotic formula for general moduli. The work relies on the study of a special divisor sum function, called $\mathcal{D}_q$-function, which plays a key role in deducing the main terms. Another key ingredient is an application of new bounds for double sums in Kloosterman sums, applied in the estimate of the error terms.