The fourth moment of quadratic Dirichlet $L$-functions

Abstract

We study the fourth moment of quadratic Dirichlet $L$-functions at $s=\frac{1}{2}$. We show an asymptotic formula under the generalized Riemann hypothesis, and obtain a precise lower bound unconditionally. The proofs of these results follow closely arguments of Soundararajan and Young [19] and Soundararajan [17].

abstract and introduction substantially revised; typos/mistakes fixed; more detail given; accept by Math. Z