1. Introduction. One of the classical unsolved problems in analytic number theory is concerned with counting the number of lattice points that lie in a circle. If A(r) is the number of lattice points in SD(r) = {(x, y) \x + y ^ r } then the problem is to find the least value of 6 such that (1) E(r) m A{r) -rr2 = 0(r). It has been shown by Loo-Keng Hua [1] that (1) holds for 8 = |f and by G. H. Hardy [2] that it does not hold for 0 = f. A conjecture, frequently attributed to Hardy, asserts that

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... 1) is valid for all 8 > §. The availability of high-speed digital computers suggests actually evaluating the deviation E(r) for "large" values of r in the hope of determining some further evidence of its behavior. At least three independent efforts in this direction have recently been made (as far as we know in the order: [3], [4], and the present paper). It is apparent from our results that the first effort [3] employed insufficiently large radii (r < 2000) and that the second effort [4] is incorrect for r ^ 3000. The present calculations, which extend to r = 259,750, suggest that (1) should be valid for some 8 < if. However, it also seems clear that the computations which would be required to approximate any such lower estimate are impractical on an IBM 7090 or even on any other currently existing computers. A formula for computing A(r) is presented in Section 2. An efficient algorithm for evaluating this formula on a 7090 or similar machine and the corresponding program are described in Section 3. The numerical results are discussed in Section 4. 2. Formulation. For any positive real number Z, let [Z] denote the integer part of Z. For any radius r, we define the integers