High-dimensional D. H. Lehmer problem over short intervals

Let k be a positive integer and n a nonnegative integer, 0 < . 1, ..., . k+1 . 1 be real numbers and w = (. 1, . 2, ..., . k+1). Let q.max{[1.i]:1.i.k+1} be a positive integer, and a an integer coprime to q. Denote by N(a, k,w, q,n) the 2n-th moment of (b 1 ... b k . c) with b 1 ...b k c . a (mod q), 1 . b i . . i q (i = 1, ..., k), 1 . c . . k+1q and 2 . (b 1 + ... + b k + c). We first use the properties of trigonometric sum and the estimates of n-dimensional Kloosterman sum to give an interesting asymptotic formula for N(a, k,w, q, n), which generalized the result of Zhang. Then we use the properties of character sum and the estimates of Dirichlet L-function to sharpen the result of N(a, k,w, q, n) in the case of w=(12,12,.,12) and n = 0. In order to show our result is close to the best possible, the mean-square value of N(a,k,q)..k(q)2k+2 and the mean value weighted by the high-dimensional Cochrane sum are studied too.