Geodesic
A MEAN VALUE RELATED TO D. H. LEHMER'S PROBLEM AND THE RAMANUJAN'S SUM*
Glasgow mathematical journal, Tome 54 (2012) no. 1, pp. 155-162

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DOI

Let q > 1 be an odd integer and c be a fixed integer with (c, q) = 1. For each integer a with 1 ≤ a ≤ q − 1, it is clear that there exists one and only one b with 0 ≤ b ≤ q − 1 such that ab ≡ c (mod q). Let N(c, q) denotes the number of all solutions of the congruence equation ab ≡ c (mod q) for 1 ≤ a, b ≤ q − 1 in which a and b are of opposite parity, where b is defined by the congruence equation bb ≡ 1(modq). The main purpose of this paper is using the mean value theorem of Dirichlet L-functions to study the mean value properties of a summation involving (N(c, q) − φ(q)) and Ramanujan's sum, and give two exact computational formulae.
DOI : 10.1017/S0017089511000498
Mots-clés : Primary 11L40, 11F20 
WENPENG, ZHANG. A MEAN VALUE RELATED TO D. H. LEHMER'S PROBLEM AND THE RAMANUJAN'S SUM*. Glasgow mathematical journal, Tome 54 (2012) no. 1, pp. 155-162. doi: 10.1017/S0017089511000498
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