On the r-th hyper-Kloosterman sums and a problem of D. H. Lehmer
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For any integer k≥2, let P(c,k+1;q) be the number of all k+1-tuples with positive integer coordinates (a_1, a_2,…,a_{k+1}) such that 1≤a_i≤q, (a_i, q)=1,a_1 a_2…a_{k+1}≡ c(mod q) and 2│(a_1+a_2+…+a_{k+1}), and E(c,k+1;q)=P(c,k+1;q)-Φi^{k}(q)}/{2}. The main purpose of this paper is using the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of the r-th hyper-Kloosterman sums Kl(h,k+1,r;q) and E(c,k+1;q), and give an interesting mean value formula.
Abstract
For any integer k≥2, let P(c,k+1;q) be the number of all k+1-tuples with positive integer coordinates (a_1, a_2,…,a_{k+1}) such that 1≤a_i≤q, (a_i, q)=1,a_1 a_2…a_{k+1}≡ c(mod q) and 2│(a_1+a_2+…+a_{k+1}), and E(c,k+1;q)=P(c,k+1;q)-Φi^{k}(q)}/{2}. The main purpose of this paper is using the properties of Gauss sums, primitive characters and the mean value theorems of Dirichlet L-functions to study the hybrid mean value of the r-th hyper-Kloosterman sums Kl(h,k+1,r;q) and E(c,k+1;q), and give an interesting mean value formula.
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