A NEW MEAN VALUE RELATED TO D. H. LEHMER'S PROBLEM AND KLOOSTERMAN SUMS

Abstract

Abstract. 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 thereexists one and only one b with 0 ≤b ≤q −1 such that ab ≡c (mod q).Let N(c,q) denote the number of all solutions of the congruence equationab ≡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 mainpurpose of this paper is using the mean value theorem of Dirichlet L-functions to study the mean value properties of a summation involvingN(c,q) − 12 φ(q)and Kloosterman sums, and give a sharper asymptoticformula for it. 1. IntroductionLet p be an odd prime and c be a fixed integer with (c,p) = 1. For eachinteger a with 1 ≤ a ≤ p − 1, it is clear that there exists one and only one bwith 0 ≤ b ≤ p − 1 such that ab ≡ c (mod p). Let M(c,p) denote the numberof cases in which a and b are of opposite parity. In reference [4], D. H. Lehmerasked to study M(1,p) or at least to say something nontrivial about it. It isknown that M(1,p) ≡ 2 or 0 (mod 4) when p ≡ ±1 (mod 4). For generalodd number q ≥ 3, Wenpeng Zhang [6] studied the asymptotic properties ofM(1,q), and obtained a sharp asymptotic formula:M(1,q) =12φ(q) +Oq