A hybrid mean value involving hyper-Kloosterman sums and mth Cochrane sum.

Let h, b, c, and q ≥ 3 be integers, and let a ¯ satisfy the equation a a ¯ ≡ 1 mod q. The general mth Cochrane sum is defined as C m h , c , q = ∑ a = 1 ′ q c a ¯ / q ha m / q. The purpose of this paper is to study the hybrid mean value involving hyper-Kloosterman sums and the mth Cochrane sum ∑ x = 1 ′ q I m + 1 , h ; q C m h , c , q by applying the properties of Gauss sums, primitive characters, and a mean value theorem of Dirichlet L-functions. Similarly, we also have the hybrid mean value involving mth Cochrane sum and the general Kloosterman sum defined by Ye [Y.B. Ye, Hyper-Kloosterman sums and estimation of exponential sums of polynomials of higher degrees, Acta Arith., 86(3):255–267, 1998] as K m b ; c ; q : = ∑ x = 1 ′ q e bx m + c x ¯ / q. For m = 1, we obtain a better asymptotic formula than that of Zhang in [W.P. Zhang, On a Cochrane sum and its hybrid mean value formula, J. Math. Anal. Appl., 267(1):89–96, 2002]. [ABSTRACT FROM AUTHOR]