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>

On the 1-error linear complexity of two-prime generator

<abstract><p>Jing et al. dealed with all possible Whiteman generalized cyclotomic binary sequences $ s(a, b, c) $ with period $ N = pq $, where $ (a, b, c) \in \{0, 1\}^3 $ and $ p, q $ are distinct odd primes (Jing et al. arXiv:2105.10947v1, 2021). They have determined the autocorrelation distribution and the 2-adic complexity of these sequences in a unified way by using group ring language and a version of quadratic Gauss sums. In this paper, we determine the linear complexity and the 1-error linear complexity of $ s(a, b, c) $ in details by using the discrete Fourier transform (DFT). The results indicate that the linear complexity of $ s(a, b, c) $ is large enough and stable in most cases.</p></abstract>

On the classical Gauss sums and their some new identities

<abstract><p>In this paper, we use the analytic methods and the properties of the classical Gauss sums to study the calculating problems of some Gauss sums involving the character of order $ 12 $ modulo an odd prime $ p $, and obtain several new and interesting identities for them.</p></abstract>

On Gauss sums over dedekind domains

It is a difficult question to generalize Gauss sums to a ring of algebraic integers of an arbitrary algebraic number field. In this paper, we define and discuss Gauss sums over a Dedekind domain of finite norm. In particular, we give a Davenport–Hasse type formula for some special Gauss sums. As an application, we give some more precise formulas for Gauss sums over the algebraic integer ring of an algebraic number field (see Theorems 4.1 and 4.2).

The mean value of the generalized Dirichlet L-functions with the weight of the Gauss sums

Let [Formula: see text] be an integer, and let [Formula: see text] denote a Dirichlet character modulo [Formula: see text]. For any real number [Formula: see text], we define the generalized Dirichlet [Formula: see text]-function as [Formula: see text] where [Formula: see text] with [Formula: see text] and [Formula: see text] both real. It can be extended to all [Formula: see text] using analytic continuation. For any integer [Formula: see text], the famous Gauss sum [Formula: see text] is defined as [Formula: see text] where [Formula: see text]. This paper uses analytic methods to study the mean value properties of the generalized Dirichlet [Formula: see text]-functions with the weight of the Gauss sums, and a sharp asymptotic formula is obtained.

A Hybrid Power Mean Involving the Dedekind Sums and Cubic Gauss Sums

The main purpose of this paper is using analytic methods and the properties of the Dedekind sums to study one kind hybrid power mean calculating problem involving the Dedekind sums and cubic Gauss sum and give some interesting calculating formulae for it.

On the Hybrid Power Mean Involving the Character Sums and Dedekind Sums

The main purpose of this paper is to use the elementary and analytic methods, the properties of Gauss sums, and character sums to study the computational problem of a certain hybrid power mean involving the Dedekind sums and a character sum analogous to Kloosterman sum and give two interesting identities for them.

One Kind Special Gauss Sums and their Mean Square Values

In this paper, we introduce one kind special Gauss sums; then, using the elementary and analytic methods to study the mean value properties of these kind sums, we obtain several exact calculating formulae for them.