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

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