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 integral P-infinity(0)2(x)e(-x/T)dx = 1/4(T/pi)(3/2) Sigma (infinity)(n=1)r(2)(n)n(-3/2)-T+O-epsilon(T2/3+epsilon), improving the earlier result with exponent 5/6 in the error term. The new bound is obtained by using results of F. Chamizo on the correlated sum Sigma (n less than or equal tox)r(n)r(n + h), where r(n) is the number of representations of n as a sum of two integer squares.