Some Identities Involving Certain Hardy Sums and General Kloosterman Sums

Abstract

Using the properties of Gauss sums, the orthogonality relation of character sum and the mean value of Dirichlet L-function, we obtain some exact computational formulas for the hybrid mean value involving general Kloosterman sums K ( r , l , λ ; p ) and certain Hardy sums S 1 ( h , q ) ∑ m = 1 p − 1 ∑ s = 1 p − 1 K ( m , n , λ ; p ) K ( s , t , λ ; p ) S 1 ( 2 m s ¯ , p ) , ∑ m = 1 p − 1 ∑ s = 1 p − 1 | K ( m , n , λ ; p ) | 2 | K ( s , t , λ ; p ) | 2 S 1 ( 2 m s ¯ , p ) . Our results not only cover the previous results, but also contain something quite new. Actually the previous authors just consider the case of the principal character λ modulo p, while we consider all the cases.

Bazı Hardy Toplamları ve Ramanujan Toplamının Ortalama Değeri Hakkında Yeni Bağıntılar

“…Regarding the properties of Hardy sums and other celebrated sums such as Ramanujan sum and Kloosterman sum, some authors [7,8,11,[17][18][19] focused on them and obtained meaningful and interesting outcomes.…”