A note on the Lehmer problem over short intervals |
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Authors: | Ya Ming Lu Yuan Yi |
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Institution: | 1.School of Science,Xi’an Jiaotong University,Xi’an,P. R. China |
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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
c=1
p−1(r(N, c; p) − 1/2N)2. |
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Keywords: | Lehmer problem character sum Dirichlet L-function |
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