On the Generalization of Lehmer Problem and High-Dimension Kloosterman Sums

Abstract

For any fixed integerk≥2and integerrwithr, p=1, it is clear that there existkintegers1≤ai≤p-1 i=1, 2, …, ksuch thata1a2⋯ak≡r mod p. LetN(k,r;p)denote the number of alla1, a2, ⋯aksuch thata1a2⋯ak≡r mod pand 2†a1+a2+⋯ + ak. In this paper, we will use the analytic method and the estimate for high-dimension Kloosterman sums to study the asymptotic properties ofN(k,r;p)and give two interesting asymptotic formulae for it.