On the number of coprime integer pairs within a circle
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On the number of coprime integer pairs within a circle
On the number of coprime integer pairs within a circle by
Wenguang Zhai (Jinan) and Xiaodong Cao (Beijing) 1. Introduction. Let P (x) denote the number of integer pairs within the circle a2 + b2 ≤ x, and E(x) denote the difference P (x) − πx. Then the well-known circle problem is to estimate the upper bound of E(x) and the best result at present is (1.1)
E(x) = O(x23/73+ε ).
See Huxley [6]. Let V (x) denote the number of coprime integer pairs within the circle a2 + b2 ≤ x. It is an exercise to deduce that 6 (1.2) V (x) = x + O(x1/2 exp(−c log3/5 x(log log x)−2/5 )), π where c is some absolute constant. The problem of reducing the exponent 1/2 is open. One way to make progress is to assume the Riemann Hypothesis (RH). W. G. Nowak [11] proved that RH implies 6 (1.3) V (x) = x + O(x15/38+ε ). π D. Hensley [5] also got a result of this type, but with a larger exponent. The aim of this paper is to further improve this result. We have the following Theorem. If RH is true, then 6 (1.4) V (x) = x + O(x11/30+ε ). π Notations. e(x) = exp(2πix). m ∼ M means c1 M ≤ m ≤ c2 M for absolute constants c1 and c2 . E(x) always denotes the error term in the circle problem. ε denotes an arbitrary small positive number and may be different at each occurrence. 1991 Mathematics Subject Classification: 11N37, 11P21. This work is supported by Natural Science Foundation of Shandong Province (Grant No. Q98A02110). [1]
2
W. G. Zhai and X. D. Cao
The authors thank Professor W. G. Nowak for kindly sending reprints of some of his papers. 2. Some preliminary lemmas and results. The following lemmas are needed. Lemma 1. Suppose 0 < c1 λ1 ≤ |f 0 (n)| ≤ c2 λ1 and |f 00 (n)| ∼ λ1 N −1 for N ≤ n ≤ cN . Then X 1/2 1/2 e(f (n)) λ−1 . 1 + λ1 N N
