高斯圆问题中二阶矩的拉普拉斯变换
The Laplace Transform of the Second Moment in the Gauss Circle Problem
摘要：
高斯圆问题涉及半径为$\sqrt{n}$的圆的面积与其包含的格点的数目之间的差$P_2(n)$。研究了系数为$P_2(n)^2$的Dirichlet级数，证明了该级数对$\mathbb{C}$具有亚纯延拓.使用这个系列，我们证明了拉普拉斯变换的$P_2(n)^2$满足$\int_0^\infty P_2(t)^2 e^{-t/X} \, dt = C X^{3/2} -X + O(X^{1/2+ε})$，这给出了一个省电的改进，以前的结果的Ivic [Ivic 1996]。同样，我们研究了亚纯连续的Dirichlet级数的相关性$r_2(n+h)r_2(n)$，其中$h$是固定的和$r_2(n)$表示的$n$作为两个平方和的表示数。我们使用这个狄利克雷级数证明$\sum_{n \geq 1} r_2(n+h)r_2(n) e^{-n/X}$的渐近性，并提供一个额外的评估领先的系数在渐近$\sum_{n \leq X} r_2(n+h)r_2(n)$。
英文摘要：
The Gauss circle problem concerns the difference $P_2(n)$ between the area of a circle of radius $\sqrt{n}$ and the number of lattice points it contains. In this paper, we study the Dirichlet series with coefficients $P_2(n)^2$, and prove that this series has meromorphic continuation to $\mathbb{C}$. Using this series, we prove that the Laplace transform of $P_2(n)^2$ satisfies $\int_0^\infty P_2(t)^2 e^{-t/X} \, dt = C X^{3/2} -X + O(X^{1/2+ε})$, which gives a power-savings improvement to a previous result of Ivic [Ivic1996]. Similarly, we study the meromorphic continuation of the Dirichlet series associated to the correlations $r_2(n+h)r_2(n)$, where $h$ is fixed and $r_2(n)$ denotes the number of representations of $n$ as a sum of two squares. We use this Dirichlet series to prove asymptotics for $\sum_{n \geq 1} r_2(n+h)r_2(n) e^{-n/X}$, and to provide an additional evaluation of the leading coefficient in the asymptotic for $\sum_{n \leq X} r_2(n+h)r_2(n)$.
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