Displaying similar documents to “Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums”

On the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum

Yongguang Du, Huaning Liu (2013)

Czechoslovak Mathematical Journal

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The main purpose of the paper is to study, using the analytic method and the property of the Ramanujan’s sum, the computational problem of the mean value of the mixed exponential sums with Dirichlet characters and general Gauss sum. For integers m , n , k , q , with k ≥ 1 and q ≥ 3 , and Dirichlet characters χ , χ ¯ modulo q we define a mixed exponential sum C ( m , n ; k ; χ ; χ ¯ ; q ) = ∑ a = 1 q w i d t h 0 p t h e i g h t 1 e m ' χ ( a ) G k ( a , χ ¯ ) e m a k + n a k ¯ q , with Dirichlet character χ and general Gauss sum G k ( a , χ ¯ ) as coefficient, where ∑ ' denotes the summation over all a such that ( a , q ) = 1 , a a ¯ ≡ 1 mod q and e ( y ) = e 2 π i y . We mean value...

Two identities related to Dirichlet character of polynomials

Weili Yao, Wenpeng Zhang (2013)

Czechoslovak Mathematical Journal

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Let q be a positive integer, χ denote any Dirichlet character mod q . For any integer m with ( m , q ) = 1 , we define a sum C ( χ , k , m ; q ) analogous to high-dimensional Kloosterman sums as follows: C ( χ , k , m ; q ) = ∑ a 1 = 1 q ' ∑ a 2 = 1 q ' ⋯ ∑ a k = 1 q ' χ ( a 1 + a 2 + ⋯ + a k + m a 1 a 2 ⋯ a k ¯ ) , where a · a ¯ ≡ 1 mod q . The main purpose of this paper is to use elementary methods and properties of Gauss sums to study the computational problem of the absolute value | C ( χ , k , m ; q ) | , and give two interesting identities for it.