论文标题
高斯领域中二次Hecke $ l $ functions的中心价值的第一时刻
First moment of central values of quadratic Hecke $L$-functions in the Gaussian field
论文作者
论文摘要
我们使用Double Dirichlet系列的方法评估了高斯领域中Qudratic Hecke $ l $ functions的中心价值的平滑第一瞬间。我们获得的渐近公式在广义的Riemann假设下具有大小$ o（x^{1/4+\ varepsilon}）$的误差项。同样的方法还允许我们获得所有$ x $，$ y $的渐近公式，用于平滑的双字符总和，涉及$ \ sum_ {n（m）\ leq x，n（n）\ leq y} \ left（\ frac {m} {m} {n} {n} {n} {n} {n} {n} {n} \ right）$ \ weft $ \ frac $ n weft（高斯领域中的二次符号。
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.
