On s-dimensional incomplete Kloosterman sums

In this paper we utilize the estimation of number of solutions of congruence to obtain the upper bound of incomplete Kloosterman sums, which improves the Shparlinski's result and removes the parameter r.     

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     

On character sums over a short interval

The main purpose of this paper is using the analytic methods to study the hybrid mean value involving the character sums, general quadratic Gauss sums and general Kloosterman sums, and give several interesting mean value formulae.     

Some applications of smooth bilinear forms with Kloosterman sums

We revisit a recent bound of I. Shparlinski and T. Zhang on bilinear forms with Kloosterman sums, and prove an extension for correlation sums of Kloosterman sums against Fourier coefficients of modular forms. We use these bounds to improve on earlier results on sums of Kloosterman sums along the primes and on the error term of the fourth moment of Dirichlet L-functions.     

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.     

Generalized Cochrane sums and Cochrane-Hardy sums

In this paper, the generalized Cochrane sums and Cochrane-Hardy sums are defined. The arithmetic properties of the generalized Cochrane sums are studied, and the Cochrane-Hardy sums are expressed in terms of the generalized Cochrane sums. Analogues of Subrahmanyam's identity and Knopp's theorem are given and proved. Finally, the hybrid mean value of generalized Cochrane sums, Cochrane-Hardy sums and Kloosterman sums is studied, and a few asymptotic formulae are obtained.     

Reducing character sums to Kloosterman sums

We apply a bound for very short Kloosterman type sums, due to A. A. Karatsuba, to deduce a bound for a mean-value of short sums of Dirichlet characters.     

On zeros of Kloosterman sums

A Kloosterman zero is a non-zero element of ${{\mathbb F}_q}$ for which the Kloosterman sum on ${{\mathbb F}_q}$ attains the value 0. Kloosterman zeros can be used to construct monomial hyperbent (bent) functions in even (odd) characteristic, respectively. We give an elementary proof of the fact that for characteristic 2 and 3, no Kloosterman zero in ${{\mathbb F}_q}$ belongs to a proper subfield of ${{\mathbb F}_q}$ with one exception that occurs at q = 16. It was recently proved that no Kloosterman zero exists in a field of characteristic greater than 3. We also characterize those binary Kloosterman sums that are divisible by 16 as well as those ternary Kloosterman sums that are divisible by 9. Hence we provide necessary conditions that Kloosterman zeros must satisfy.     

Exponential Sums and Coding Theory: A Review

This article is a survey of several recent applications of methods from analytic number theory to research in coding theory, including results on Kloosterman codes, binary Goppa codes, and prime phase shift sequences. The mathematical methods focus on exponential sums, in particular Kloosterman sums. The interrelationships with the Weil–Carlitz–Uchiyama bound, results on Hecke operators, theorems of Bombieri and Deligne and the Eichler–Selberg trace formula are reviewed.     

Power-moments of SL3(ℤ) Kloosterman sums

Classical Kloosterman sums have a prominent role in the study of automorphic forms on GL₂ and further they have numerous applications in analytic number theory. In recent years, various problems in analytic theory of automorphic forms on GL₃ have been considered, in which analogous GL₃-Kloosterman sums (related to the corresponding Bruhat decomposition) appear. In this note we investigate the first four power-moments of the Kloosterman sums associated with the group SL₃(?). We give formulas for the first three moments and a nontrivial bound for the fourth.     

Congruences with intervals and arbitrary sets

Given a prime p, an integer $$H\in [1,p)$$, and an arbitrary set $${\mathcal {M}} \subseteq {\mathbb {F}} _p^*$$, where $${\mathbb {F}} _p$$ is the finite field with p elements, let $$J(H,{\mathcal {M}} )$$ denote the number of solutions to the congruence $$\begin{aligned} xm\equiv yn~\mathrm{mod}~ p \end{aligned}$$for which $$x,y\in [1,H]$$ and $$m,n\in {\mathcal {M}} $$. In this paper, we bound $$J(H,{\mathcal {M}} )$$ in terms of p, H, and the cardinality of $${\mathcal {M}} $$. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results of Kowalski et al. (Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums, 2018, arXiv:1802.09849).     

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.     

Bounds of trilinear sums with Kloosterman fractions

Archiv der Mathematik - We obtain a new bound for trilinear exponential sums with Kloosterman fractions which in some ranges of parameters improves that of S. Bettin and V. Chandee (2018). We also...     

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.     

A Kuznetsov Formula for Kloosterman Sums on GLn

We prove a Kuznetsov sum formula for Kloosterman sums on GL _n corresponding to the big Bruhat cell. Using this formula, a weighted sum of Kloosterman sums can be expressed in spectral decomposition. A non-trivial estimate of the spectral side might lead to a proof of cancellations in sums of Kloosterman sums on GL _n.     

An identity involving Dedekind sums and generalized Kloosterman sums

The various properties of classical Dedekind sums S(h, q) have been investi-gated by many authors. For example, Yanni Liu and Wenpeng Zhang: A hybrid mean value related to the Dedekind sums and Kloosterman sums, Acta Mathematica Sinica, 27 (2011), 435–440 studied the hybrid mean value properties involving Dedekind sums and generalized Kloosterman sums K(m, n, r; q). The main purpose of this paper, is using the analytic methods and the properties of character sums, to study the computational problem of one kind of hybrid mean value involving Dedekind sums and generalized Kloosterman sums, and give an interesting identity.     

On Z4-Linear Goethals Codes and Kloosterman Sums

Studying the coset weight distributions of the Z₄-linear Goethals codes, e connect these codes with the Kloosterman sums. From one side, e obtain for some cases, of the cosets of weight four, the exact expressions for the number of code ords of weight four in terms of the Kloosterman sums. From the other side, e obtain some limitations for the possible values of the Kloosterman sums, hich improve the well known results due to Lachaud and Wolfmann kn:lac.     

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.     

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.