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.
Categories: math.NT