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arXiv · 2207.03746

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

Peng Gao·Liangyi Zhao
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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.

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BibTeXRIS

Peng Gao, Liangyi Zhao. 2022-07-08. First moment of central values of quadratic Hecke $L$-functions in the Gaussian field. https://doi.org/10.1142/s1793042123500793

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