Gauss Sums Research Articles

One-Kind Hybrid Power Means of the Two-Term Exponential Sums and Quartic Gauss Sums

The main purpose of this article is using the analytic methods and the properties of the classical Gauss sums to study the calculating problem of the hybrid power mean of the two-term exponential sums and quartic Gauss sums and then prove two interesting linear recurrence formulas. As applications, some asymptotic formulas are obtained.

A note on two-term exponential sum and the reciprocal of the quartic Gauss sums

The main purpose of this article is by using the properties of the fourth character modulo a prime p and the analytic methods to study the calculating problem of a certain hybrid power mean involving the two-term exponential sums and the reciprocal of quartic Gauss sums, and to give some interesting calculating formulae of them.

The Recursive Properties of the Error Term of the Fourth Power Mean of the Generalized Cubic Gauss Sums

In this paper, we use the analytic methods and the properties of the classical Gauss sums to study the properties of the error term of the fourth power mean of the generalized cubic Gauss sums and give two recurrence formulae for it.

A Four-Order Linear Recurrence Formula Involving the Quartic Gauss Sums and One Kind Two-Term Exponential Sums

The main purpose of this paper is using the analytic method and the properties of the classical Gauss sums to study the computational problem of one kind hybrid power mean involving the quartic Gauss sums and two-term exponential sums and give an interesting four-order linear recurrence formula for it. As an application, we can obtain all values of this kind hybrid power mean with mathematica software.

A Hybrid Mean Value Involving Dedekind Sums and the Generalized Kloosterman Sums

In this paper, we use the mean value theorem of Dirichlet L -functions and the properties of Gauss sums and Dedekind sums to study the hybrid mean value problem involving Dedekind sums and the general Kloosterman sums and give an interesting identity for it.

Estimations on some hybrid exponential sums related to Kloosterman sums

In this paper, we revisit the bounds of the mixed exponential sums introduced by Lv and Zhang (2020). Moreover, we give some estimations for some new hybrid exponential sums related to Kloosterman sums over finite fields of odd characteristic by using the properties of Jacobi sums and Gaussian sums.

Gaussian sums, hyper Eisenstein sums and Jacobi sums over a local ring and their applications

It is well known that any finite commutative ring is isomorphic to a direct product of local rings via the Chinese remainder theorem. Hence, there is a great significance to the study of character sums over local rings. Character sums over finite rings have applications that are analogous to the applications of character sums over finite fields. In particular, character sums over local rings have many applications in algebraic coding theory. In this paper, we firstly present an explicit description on additive characters and multiplicative characters over a certain local ring. Then we study Gaussian sums, hyper Eisenstein sums and Jacobi sums over a certain local ring and explore their properties. It is worth mentioning that we are the first to define Eisenstein sums and Jacobi sums over this local ring. Moreover, we present a connection between hyper Eisenstein sums over this local ring and Gaussian sums over finite fields, which allows us to give the absolute value of hyper Eisenstein sums over this local ring. As an application, several classes of codebooks with new parameters are presented.

Character sums over a non-chain ring and their applications

Some valuable results over rings have a promising utilization in coding theory and error-correcting code theory. In this paper, we study character sums over a certain non-chain ring and their applications in codebooks. There are two major ingredients in this study. The first ingredient is to investigate Gaussian sums, hyper Eisenstein sums, Jacobi sums over a certain non-chain ring and study the properties of these character sums. For their applications, the second ingredient is to present three classes of asymptotically optimal codebooks with respect to the Welch bound and a family of optimal codebooks with respect to the Levenshtein bound, which are constructed from character sums over a certain non-chain ring.

The character sum of polynomials with k variables and two-term exponential sums

The main purpose of this paper is using the properties of the classical Gauss sums and the analytic methods to study the computational problem of one kind of hybrid power mean involving the character sums of polynomials with k variables and the two-term exponential sums, and give an identity and asymptotic formula for it.

Local densities of diagonal integral ternary quadratic forms at odd primes

We give formulas for local densities of diagonal integral ternary quadratic forms at odd primes. Exponential sums and quadratic Gauss sums are used to obtain these formulas. These formulas (along with 2-adic densities and Siegel’s mass formula) can be used to compute the representation numbers of certain ternary quadratic forms.

Classification of Spherical Fusion Categories of Frobenius–Schur Exponent 2

In this paper, we propose a new approach towards the classification of spherical fusion categories by their Frobenius–Schur exponents. We classify spherical fusion categories of Frobenius–Schur exponent 2 up to monoidal equivalence. We also classify modular categories of Frobenius–Schur exponent 2 up to braided monoidal equivalence. It turns out that the Gauss sum is a complete invariant for modular categories of Frobenius–Schur exponent 2. This result can be viewed as a categorical analog of Arf's theorem on the classification of non-degenerate quadratic forms over fields of characteristic 2.

Base change for ramified unitary groups: The strongly ramified case

Abstract We compute a special case of base change of certain supercuspidal representations from a ramified unitary group to a general linear group, both defined over a p-adic field of odd residual characteristic. In this special case, we require the given supercuspidal representation to contain a skew maximal simple stratum, and the field datum of this stratum to be of maximal degree, tamely ramified over the base field, and quadratic ramified over its subfield fixed by the Galois involution that defines the unitary group. The base change of this supercuspidal representation is described by a canonical lifting of its underlying simple character, together with the base change of the level-zero component of its inducing cuspidal type, modified by a sign attached to a quadratic Gauss sum defined by the internal structure of the simple character. To obtain this result, we study the reducibility points of a parabolic induction and the corresponding module over the affine Hecke algebra, defined by the covering type over the product of types of the given supercuspidal representation and of a candidate of its base change.

Some identities involving Gauss sums

<abstract><p>We calculate several identities involving some Gauss sums of the $ 2^k $-order character modulo an odd prime $ p $ by using the elementary and analytic methods, and finally give several exact and interesting formulae for them. The properties of the classical Gauss sums play an important role in the proof of this paper.</p></abstract>

Gaussian sums and their application to the proof of quadraticreciprocity law

Впервые публикуется текст дипломной работы И. М. Виноградова, выполненной им под руководством Я. В. Успенского на математическом отделении физико-математического факультета Петербургского университета в 1914 г.

On the generalized quadratic Gauss sums and its upper bound estimate

On the generalized quadratic Gauss sums and its upper bound estimate

On the p ‐Adic Complexity of the Ding‐Helleseth‐Martinsen Binary Sequences

The purpose of this paper is to determine the p-adic complexity of the Ding-Helleseth-Martinsen (DHM) sequences with period N = 2q, where q = 5 (mod 8) is a prime number. We firstly use the p-adic exponential valuation, cyclotomic numbers of order four, “Gauss periods” and “quadratic Gauss sums” on finite field 𝔽q, and valued in ℤN, to determine the p-adic complexity of the DHM sequences.

On the character sums analogous to high dimensional Kloosterman sums

<abstract><p>The main purpose of this paper is using the properties of the classical Gauss sums and the analytic methods to study the computational problem of one kind of character sums analogous to high dimensional Kloosterman sums, and give some interesting identities for it.</p></abstract>

The generalized quadratic Gauss sums and its sixth power mean

<abstract><p>In this article, we using elementary methods, the number of the solutions of some congruence equations and the properties of the Legendre's symbol to study the computational problem of the sixth power mean of a certain generalized quadratic Gauss sums, and to give an exact calculating formula for it.</p></abstract>

Higher order moments of generalized quadratic Gauss sums weighted by $L$-functions

The main purpose of this paper is to study higher order moments of the generalized quadratic Gauss sums weighted by $L$-functions using estimates for character sums and analytic methods. We find asymptotic formulas for three character sums which arise naturally in the study of higher order moments of the generalized quadratic Gauss sums. We then use these character sum estimates to find asymptotic formulas for the $\nth{6}$ and $\nth{8}$ order moments of the generalized quadratic Gauss sums weighted by $L$-functions. Our asymptotic formulas satisfy a conjecture of Wenpeng Zhang.

Converse theorem of Gauss sums

In this paper, we investigate an inverse problem on the Gauss sums of characters of finite fields. Namely, given a nontrivial additive character ψ, for two multiplicative characters χ1 and χ2 of Fqn×, if the Gauss sums G(χ1⋅η,ψ)=G(χ2⋅η,ψ) for all characters η of Fq×, when is χ1 equal to χ2 up to a Frobenius twist? This paper proves that the answer is positive for regular characters when n≤5, or for n<q−12q+1 in the appendix by Zhiwei Yun. In addition, we conjectured that the answer is positive when n is prime.