Abstract
Let p be an odd prime, c be an integer with (c, p) = 1, and let N be a positive integer with N ≤ p − 1. Denote by r(N, c; p) the number of integers a satisfying 1 ≤ a ≤ N and 2 ∤ a + ā, where ā is an integer with 1 ≤ ā ≤ p − 1, aā ≡ c (mod p). It is well known that r(N, c; p) = 1/2N + O(p 1/2log2 p). The main purpose of this paper is to give an asymptotic formula for Σ p−1 c=1 (r(N, c; p) − 1/2N)2.
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Supported by National Natural Science Foundation of China (Grant No. 10601039)
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Lu, Y.M., Yi, Y. A note on the Lehmer problem over short intervals. Acta. Math. Sin.-English Ser. 27, 1115–1120 (2011). https://doi.org/10.1007/s10114-011-8554-8
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DOI: https://doi.org/10.1007/s10114-011-8554-8