The fourth moment of quadratic Dirichlet $L$-functions

Abstract

We study the fourth moment of quadratic Dirichlet L-functions at s = 1 2 . We show an asymptotic formula under the generalized Riemann hypothesis, and obtain a precise lower bound unconditionally.For simplicity, we only focus on quadratic characters of the form χ 8d , and the result for χ d may be established similarly. We note that Soundararajan and Young [18] have established the

Moments and Non-vanishing of central values of Quadratic Hecke $L$-functions in the Gaussian Field

“…In the same paper [14], Soundrarajan also obtained the third moment of the family of primitive quadratic Dirichlet L-functions. Under the assumption of the Generalized Riemann Hypothesis (GRH), Q. Shen obtained an asymptotic formula in [12] for the fourth moment of the same family. Analogue to quadratic twists by Dirichlet characters, there is also an intensive study on moments of various families of modular forms.…”

Secondary terms in the asymptotics of moments of L-functions

“…In 1981, Jutila [25] computed the first and second moment, and Soundararajan [36], [12], and Young [42] obtained asymptotics for the third moment with different error terms. Under the generalized Riemann hypothesis, Shen [35] obtained an asymptotic formula for the fourth moment (using the method of Soundararajan and Young [37]). It was conjectured [10] that…”

First Moments of Some Hecke $L$-functions of Prime Moduli

“…A valid asymptotic formula for the third moment of central values of the family of quadratic Dirichlet L-functions was first established by K. Soundararajan in [26] and the error term for a smoothed version was later improved by M. P. Young in [30]. Recently, Q. Shen obtained a valid asymptotic formula in [24] for the fourth moment of central values of the same family of L-functions under the Generalized Riemann Hypothesis (GRH).…”