Abstract
Let k be a positive integer and n a nonnegative integer, 0 < λ 1, ..., λ k+1 ≤ 1 be real numbers and w = (λ 1, λ 2, ..., λ k+1). Let $$q \geqslant \max \left\{ {\left[ {\frac{1} {{\lambda _i }}} \right]:1 \leqslant i \leqslant k + 1} \right\}$$ 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 ≤ λ i q (i = 1, ..., k), 1 ≤ c ≤ λ 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 $$w = \left( {\tfrac{1} {2},\tfrac{1} {2}, \ldots ,\tfrac{1} {2}} \right)$$ and n = 0. In order to show our result is close to the best possible, the mean-square value of $$N\left( {a,k,q} \right) - \tfrac{{\varphi ^k (q)}} {{2^{k + 2} }}$$ and the mean value weighted by the high-dimensional Cochrane sum are studied too.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.