# On a Problem of D. H. Lehmer and Kloosterman Sums

DOI: 10.1007/s00605-002-0529-5

- Cite this article as:
- Wenpeng, Z. Monatsh. Math. (2003) 139: 247. doi:10.1007/s00605-002-0529-5

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## Abstract.

Let *q* ⩾ 3 be an odd number, *a* be any fixed positive integer with (*a*, *q*) = 1. For each integer *b* with 1 ⩽ *b* < *q* and (*b*, *q*) = 1, it is clear that there exists one and only one *c* with 0 < *c* < *q* such that *bc* ≡ *a* (mod *q*). Let *N*(*a*, *q*) denote the number of all solutions of the congruent equation *bc* ≡ *a* (mod *q*) for 1 ⩽ *b*, *c* < *q* in which *b* and *c* are of opposite parity, and let \(E(a, q)=N(a, q)-{1\over 2}\phi (q)\). The main purpose of this paper is to study the distribution properties of *E*(*a*, *q*), and to give a sharper hybrid mean value formula involving *E*(*a*, *q*) and Kloosterman sums.