关于圆问题中正方形的拉普拉斯变换的一个注记
A note on the Laplace transform of the square in the circle problem
摘要：
如果$P(x)$是圆问题中的误差项，则证明了$$\int_0^\infty P^2(x)e^{-x/T}dx = {1\over4}({T\overπ})^{3/2} \sum_{n=1}^\infty r^2(n)n^{-3/2} - T + O_ε(T^{2/3+ε}), $$改进了作者早先的指数5/6.利用F. Chamizo对相关和$\sum_{n\le x}r(n)r(n+h)$，其中$r(n)$是$n$表示为两个平方和的数量。
英文摘要：
If $P(x)$ is the error term in the circle problem, then it is proved that $$\int_0^\infty P^2(x)e^{-x/T}dx = {1\over4}({T\overπ})^{3/2} \sum_{n=1}^\infty r^2(n)n^{-3/2} - T + O_ε(T^{2/3+ε}), $$ improving the author's earlier exponent 5/6. The new bound is obtained by using results of F. Chamizo on the correlated sum $\sum_{n\le x}r(n)r(n+h)$, where $r(n)$ is the number of representations of $n$ as a sum of two squares.
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