Kloosterman Sums and a Problem of D.H.Lehmer

A classical problem of D. H. Lehmer suggests the study of distributions of elements of Z/pZ of opposite parity to the multiplicative inverse mod p. Zhang initiated this problem and found an asymptotic evaluation of the number of such elements. In this paper, an asymptotic formula for the fourth moment of the error term of Zhang is proved,from which one may see that Zhang's error term is optimal up to the logarithm factor.The method also applies to the case of arbitrary positive integral moments.