Kloosterman double sums

In this paper estimates of incomplete Kloosterman double sums with weights are obtained. Translated fromMatematicheskie Zametki, Vol. 66, No. 5, pp. 682–687, November, 1999.     

On the moments of Kloosterman sums and fibre products of Kloosterman curves

Let q=p^r with p=3 and r2. We give a recursion formula for the moments of a Kloosterman sum over the finite field , which utilizes known weight formulae for the ternary Melas code M of length q−1. The method is illustrated by giving explicit formulae for the moments up to the tenth moment. As an application for the formulae, and for their analogues obtained earlier in case p=2, we get the exact number of rational points on fibre products of certain Kloosterman curves. As a corollary we obtain identities between Ramanujan's tau-function, Kronecker class numbers, and Dickson polynomials.     

On ternary Kloosterman sums modulo 12

Let K(a) denote the Kloosterman sum on . It is easy to see that for all . We completely characterize those for which , and . The simplicity of the characterization allows us to count the number of the belonging to each of these three classes. As a byproduct we offer an alternative proof for a new class of quasi-perfect ternary linear codes recently presented by Danev and Dodunekov.     

Kloosterman和的角度

有限域 Ｆｑ（ｑ为奇）上的 Ｋｌｏｏｓｔｅｒｍａｎ和是两个模为 ｑ１／２的共轭复数之和．这个复数的角度就称为相应的Ｋｌｏｏｓｔｅｒｍａｎ和的角度．我们在本文给出了Ｋｌｏｏｓｔｅｒｍａｎ和的角度的一些结果,改进了Ｋａｔｚ Ｎ．［１］的一些结果,也改进和推广了Ｃｏｎｒｅｙ　Ｊ．和Ｉｗａｎｉｅ Ｈ．［２］的结果．     

Divisibility properties of classical binary Kloosterman sums

Let K(a) be the so-called classical Kloosterman sum over . In this paper, we compute K(a) modulo 24 for even m, completing our previous results for odd m. We extensively study the links between K(a) and other exponential sums, especially the cubic sums. We point out (as we did for odd m) that the values K(a) are involved in the computation of the weight distributions of cosets of primitive narrow sense extended BCH codes of length 2^m and minimum distance 8. We also complete some recent results on K(a)−1 modulo 3.     

A hybrid mean value related to the Dedekind sums and Kloosterman sums

The main purpose of this paper is to use the properties of character sum and the analytic method to study a hybrid mean value problem related to the Dedekind sums and Kloosterman sums, and give some interesting mean value formulae and identities for it.     

A transform property of Kloosterman sums

An expression for the number of times a certain trace function associated with a Kloosterman sum on an extension field assumes a given value in the base field is given and its properties explored. The relationship of this result to the enumeration of certain types of irreducible polynomials over fields of characteristic two or three and to the weights in the dual of a Melas code is considered. It is argued that the expressions obtained for the trace functions, while simply related to the Kloosterman sums, can be more directly useful than the exponential sums themselves in certain applications. In addition, they enjoy properties that are of independent interest.     

Mean value of some exponential sums and applications to Kloosterman sums

Let q, m, n, k be integers with q?3 and k?1, define the exponential sum

     

On Cochrane sums over short intervals

The main purpose of this paper is to study the mean square value problem of Cochrane sums over short intervals by using the properties of Gauss sums and Kloosterman sums, and finally give a sharp asymptotic formula.     

Seventh power moments of Kloosterman sums

Evaluations of the n-th power moments S _n of Kloosterman sums are known only for n ⩽ 6. We present here substantial evidence for an evaluation of S ₇ in terms of Hecke eigenvalues for a weight 3 newform on Γ_O(525) with quartic nebentypus of conductor 105. We also prove some congruences modulo 3, 5 and 7 for the closely related quantity T ₇, where T _n is a sum of traces of n-th symmetric powers of the Kloosterman sheaf.     

Mean Value of Kloosterman Sums over Short Intervals

This paper is concerned with a kind of mean value problem of Kloosterman sums, which will lead to a sum of Kloosterman sums over short intervals.     

Kloosterman sum identities and low-weight codewords in a cyclic code with two zeros

We apply relations of n-dimensional Kloosterman sums to exponential sums over finite fields to count the number of low-weight codewords in a cyclic code with two zeros. As a corollary we obtain a new proof for a result of Carlitz which relates one- and two-dimensional Kloosterman sums. In addition, we count some power sums of Kloosterman sums over certain subfields.     

Classical Kloosterman sums: Representation theory, magic squares, and Ramanujan multigraphs

We consider a certain finite group for which Kloosterman sums appear as character values. This leads us to consider a concrete family of commuting hermitian matrices which have Kloosterman sums as eigenvalues. These matrices satisfy a number of “magical” combinatorial properties and they encode various arithmetic properties of Kloosterman sums. These matrices can also be regarded as adjacency matrices for multigraphs which display Ramanujan-like behavior.     

Mean value of mixed exponential sums

For integers , , , with , and Dirichlet character , we define a mixed exponential sum

where , and denotes the summation over all with . The main purpose of this paper is to study the mean value of

and to give a related identity on the mean value of the general Kloosterman sum

where .

     

A generalization of power moments of Kloosterman sums

We find an expression for a sum which can be viewed as a generalization of power moments of Kloosterman sums studied by Kloosterman and Salié. Received: 24 March 2006     

Second moments of Dirichlet L-functions weighted by Kloosterman sums

For the general modulo q ? 3 and a general multiplicative character χ modulo q, the upper bound estimate of |S(m, n, 1, χ, q)| is a very complex and difficult problem. In most cases, the Weil type bound for |S(m, n, 1, χ, q)| is valid, but there are some counterexamples. Although the value distribution of |S(m, n, 1, χ, q)| is very complicated, it also exhibits many good distribution properties in some number theory problems. The main purpose of this paper is using the estimate for k-th Kloosterman sums and analytic method to study the asymptotic properties of the mean square value of Dirichlet L-functions weighted by Kloosterman sums, and give an interesting mean value formula for it, which extends the result in reference of W. Zhang, Y.Yi, X.He: On the 2k-th power mean of Dirichlet L-functions with the weight of general Kloosterman sums, Journal of Number Theory, 84 (2000), 199–213.