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

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

2003-12-12

arxiv.org/abs/math/0312255v1

Studia Scientiarum Mathematicarum Hungarica 37(2001), 391-399

Mathematics

Number Theory

8 pages

Scientific paper

If $P(x)$ is the error term in the circle problem, then it is proved that $$\int_0^\infty P^2(x)e^{-x/T}dx = {1\over4}({T\over\pi})^{3/2} \sum_{n=1}^\infty r^2(n)n^{-3/2} - T + O_\epsilon(T^{2/3+\epsilon}), $$ 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.

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