A note on the Laplace transform of the square in the circle problem

Abstract

If $P(x)$ is the error term in the circle problem, then it is proved that $${\int }_0^{\infty }{P}^2(x)e^{-x/T}dx=\frac{1}{4}(\frac{T}{\pi }{)}^{3/2}\sum _{n=1}^{\infty }{r}^2(n)n^{-3/2}-T+O_{\epsilon }(T^{2/3+\epsilon }),$$ 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.

8 pages

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