First moment of central values of quadratic Hecke $L$-functions in the Gaussian field

TL;DR

This paper computes the average value of quadratic Hecke L-functions at the central point in the Gaussian field using double Dirichlet series, providing asymptotic formulas with a specified error term under GRH.

Contribution

It introduces a novel application of double Dirichlet series to evaluate the first moment of quadratic Hecke L-functions in the Gaussian field, including error analysis under GRH.

Findings

Derived asymptotic formulas for the first moment of L-values

Established error bounds of size O(X^{1/4+ε}) under GRH

Extended approach to double character sums involving quadratic symbols

Abstract

We evaluate the smoothed first moment of central values of a family of qudratic Hecke $L$-functions in the Gaussian field using the method of double Dirichlet series. The asymptotic formula we obtain has an error term of size $O(X^{1/4+\varepsilon})$ under the generalized Riemann hypothesis. The same approach also allows us to obtain asymptotic formulas for all $X$, $Y$ for a smoothed double character sum involving $\sum_{N(m) \leq X, N(n)\leq Y}\left( \frac{m}{n} \right)$, where $\left( \frac{\cdot}{n} \right)$ denotes the quadratic symbol in the Gaussian field.