Infinite class of new sign ambiguities

In 1934, two kinds of multiplicative relations, the norm and the Davenport-Hasse relations, between Gauss sums, were known. In 1964, H. Hasse conjectured that the norm and the Davenport-Hasse relations were the only multiplicative relations connecting Gauss sums over Fp. However, in 1966, K. Yamamoto provided a simple counterexample disproving the conjecture. This counterexample was a new type of multiplicative relation, called a sign ambiguity, involving a ± sign not connected to elementary properties of Gauss sums. In this paper, we give an explicit product formula involving Gauss sums which generates an infinite class of new sign ambiguities, and we resolve the ambiguous sign by using Stickelberger?s theorem.     

一类阶为偶数的高斯和显式计算的注记

给出指数2情形下阶数为2l_1~(r1)l_1~(r2)的高斯和的显式计算公式.证明方法直接利用Stickelberger理想分解定理,进而结果独立于其他指数2情形高斯和的结果而成立.     

Complete solving of explicit evaluation of Gauss sums in the index 2 case

Let p be a prime number,N be a positive integer such that gcd(N,p) = 1,q = pf where f is the multiplicative order of p modulo N.Let χ be a primitive multiplicative character of order N over finite field Fq.This paper studies the problem of explicit evaluation of Gauss sums G(χ) in the "index 2 case"(i.e.[(Z/NZ):p] = 2).Firstly,the classification of the Gauss sums in the index 2 case is presented.Then,the explicit evaluation of Gauss sums G(χλ)(1 λ N-1) in the index 2 case with order N being general even integer(i.e.N = 2r·N0,where r,N0 are positive integers and N0 3 is odd) is obtained.Thus,combining with the researches before,the problem of explicit evaluation of Gauss sums in the index 2 case is completely solved.     

Explicit multiplicative relations between Gauss sums

H. Hasse conjectured that all multiplicative relations between Gauss sums essentially follow from the Davenport-Hasse product formula and the norm relation for Gauss sums. While this is known to be false, very few counterexamples, now known as sign ambiguities, have been given. Here, we provide an explicit product formula giving an infinite class of new sign ambiguities and resolve the ambiguous sign in terms of the order of the ideal class of quadratic primes.     

q-Dedekind type sums related to q-zeta function and basic L-series

The main purpose of this paper is to define new generating functions. By applying the Mellin transformation formula to these generating functions, we define q-analogue of Riemann zeta function, q-analogue Hurwitz zeta function, q-analogue Dirichlet L-function and two-variable q-L-function. In particular, by using these generating functions, we will construct new generating functions which produce q-Dedekind type sums and q-Dedekind type sums attached to Dirichlet character. We also give the relations between these sums and Dedekind sums. Furthermore, by using *-product which is given in this paper, we will give the relation between Dedekind sums and q-L function as well.     

On a theorem of Chowla

Let be a finite field with q=p^felements, where p is a prime number and f is a positive integer. For a nonprincipal multiplicative character χ and a nontrivial additive character ψ on , it is well known that Gauss sum has absolute value . In this paper, we investigate when is a root of unity.     

On linear combinations of special values of L-functions

Using a specific evaluation method for L-functions, it is shown that character values and weighted averages of Gauss sums can be well approximated by linear combinations of the algebraic parts of special values of L-functions under correct parity conditions. This is a surprising universal property of these special values since the coefficients of the resulting combinations turn out to be independent of the character but its parity.     

On the fourth power mean of the generalized quadratic Gauss sums

The main purpose of this paper is to use elementary methods and properties of the classical Gauss sums to study the computational problem of one kind of fourth power mean of the generalized quadratic Gauss sums mod q (a positive odd number), and give an exact computational formula for it.     

Representations of rank two affine Hecke algebras at roots of unity

In this paper, we will fully describe the irreducible representations of the crystallographic rank two affine Hecke algebras using algebraic and combinatorial methods, for all possible values of q. The focus is on the case when q is a root of unity of small order.     

Evaluation of the Hamming weights of a class of linear codes based on Gauss sums

Linear codes with a few weights have been widely investigated in recent years. In this paper, we mainly use Gauss sums to represent the Hamming weights of a class of q-ary linear codes under some certain conditions, where q is a power of a prime. The lower bound of its minimum Hamming distance is obtained. In some special cases, we evaluate the weight distributions of the linear codes by semi-primitive Gauss sums and obtain some one-weight, two-weight linear codes. It is quite interesting that we find new optimal codes achieving some bounds on linear codes. The linear codes in this paper can be used in secret sharing schemes, authentication codes and data storage systems.     

Moments of Generalized Quadratic Gauss Sums Weighted by L-Functions

The main purpose of this paper is using estimates for character sums and analytic methods to study the second, fourth, and sixth order moments of generalized quadratic Gauss sums weighted by L-functions. Three asymptotic formulae are obtained.     

A hybird mean value involving gauss sums and character sums

Let χ be a Dirichlet character modulo q > 2, and L(s, χ) denotes the Dirichlet L-function corresponding to χ. The main purpose of this paper is using the estimate for character sums and the analytic method to study the mean value properties of $ \frac{{L'}} {L}(1,\chi ) $ \frac{{L'}} {L}(1,\chi ) with the weight of Gauss sums and character sums, and give an interesting mean value formula for it.     

Gauss Sum of Index 4： （2） Non-cyclic Case

Assume that m ≥ 2, p is a prime number, （m,p（p - 1）） = 1,-1 not belong to 〈p〉 belong to （Z/mZ）^＊ and [（Z/mZ）^＊：〈p〉]=4.In this paper, we calculate the value of Gauss sum G（X）=∑x∈F^＊x（x）ζp^T（x） over Fq,where q=p^f,f=φ（m）/4,x is a multiplicative character of Fq and T is the trace map from Fq to Fp.Under our assumptions,G（x） belongs to the decomposition field K of p in Q（ζm） and K is an imaginary quartic abelian unmber field.When the Galois group Gal（K/Q） is cyclic,we have studied this cyclic case in anotyer paper：＂Gauss sums of index four：（1）cyclic case＂（accepted by Acta Mathematica Sinica,2003）.In this paper we deal with the non-cyclic case.     

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.     

A counting problem in ergodic theory and extrapolation for one-sided weights

The purpose of this paper is to prove that, given a dynamical system (X,M,μ, τ) and 0 < q < 1, the Lorentz spaces L^1,q(μ) satisfy the so-called Return Times Property for the Tail, contrary to what happens in the case q = 1. In fact, we consider a more general case than in previous papers since we work with a σ-finite measure μ and a transformation τ which is only Cesàro bounded. The proof uses the extrapolation theory of Rubio de Francia for one-sided weights. These results are of independent interest and can be applied to many other situations.     

Two identities related to dirichlet character of polynomials

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: , where a · ā ≡ 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.     

On some hybrid power moments of products of generalized quadratic Gauss sums and Kloosterman sums*

In this paper, we investigate hybrid power moments of generalized quadratic Gauss sums weighted with powers of Kloosterman sums and with powers of values of Dirichlet L-functions at 1. We obtain several exact formulas for prime and prime power modulus and some asymptotic formulas.     

On the mean value of dedekind sum weighted by the quadratic gauss sum

Various properties of classical Dedekind sums S(h, q) have been investigated by many authors. For example, Wenpeng Zhang, On the mean values of Dedekind sums, J. Théor. Nombres Bordx, 8 (1996), 429–442, studied the asymptotic behavior of the mean value of Dedekind sums, and H.Rademacher and E.Grosswald, Dedekind Sums, The Carus Mathematical Monographs No. 16, The Mathematical Association of America, Washington, D.C., 1972, studied the related properties. In this paper, we use the algebraic method to study the computational problem of one kind of mean value involving the classical Dedekind sum and the quadratic Gauss sum, and give several exact computational formulae for it.     

On the 2<Emphasis Type="Italic">k</Emphasis>-th Power Mean of Inversion of <Emphasis Type="Italic">L</Emphasis>-functions with the Weight of the Gauss Sum

Abstract The main purpose of this paper is to use the estimate for character sums and the method of trigonometric sums to study the 2k-th power mean of the inversion of Dirichlet L-functions with the weight of the Gauss sums, and give a sharper asymptotic formula. This work is supported by the Doctorate Foundation of Xi’an Jiaotong University     

Finite dimensional representations of the double affine Hecke algebra of rank 1

We classify the finite dimensional irreducible representations of the double affine Hecke algebra (DAHA) of type C^∨C₁ in the case when q is not a root of unity.