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

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 $O(X^{1\mathrm{/}4+\epsilon })$ 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 ${\sum }_{N(m)\le X,N(n)\le Y}\left(\frac{m}{n}\right)$, where $\left(\frac{\cdot }{n}\right)$ denotes the quadratic symbol in the Gaussian field.