Abstract
Let q > 2 be an odd number. For each integer a with 1 ≤ a ≤ q and (a, q) = 1, there exists one and only one [Formula: see text] with [Formula: see text] and [Formula: see text] such that [Formula: see text]. A Lehmer number is defined to be any integer a with [Formula: see text]. Let a = (a1, …, ak+1), b = (b1, …, bk+1) with 0 ≤ bi < ai for i = 1, …, k + 1, (a1 ⋯ ak+1, q) = 1, t = (t1, …, tk) satisfying 0 < t1, …, tk ≤ 1. Define [Formula: see text] The main purpose of this paper is to study the properties of N(a, b, t; q), and give a sharp asymptotic formula, by using the estimate for hyper-Kloosterman sums and properties of trigonometric sums.
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