Quadratic Gauss Sums Research Articles

ON THE 2k-TH POWER MEAN VALUE OF THE GENERALIZED QUADRATIC GAUSS SUMS

Abstract. The main purpose of this paper is using the elementary andanalytic methods to study the properties of the 2 k -th power mean valueof the generalized quadratic Gauss sums, and give two exact mean valueformulae for k = 3 and 4. 1. IntroductionLet q  2 be an integer, ˜ denotes a Dirichlet character modulo q . For anyinteger n , we de ne the generalized quadratic Gauss sums G ( n;˜ ; q ) as follows: G ( n;˜ ; q ) =∑ qa =1 ˜ ( a ) e ( na 2 q ) ; where e ( y ) = e 2 ˇiy . This sum is important, because it is a generalization ofthe classical quadratic Gauss sums G ( n;q ), which is de ned by G ( n ; q ) =∑ qa =1 e ( na 2 q ) : About the properties of G ( n;˜ ; q ), some authors had studied it, and obtainedmany interesting results. For example, for any integer n with ( n;q ) = 1, fromthe general result of Cochrane and Zheng [2] we can deduce that jG ( n;˜ ; q ) j  2 ! ( q ) q 12 ; where ! ( q ) denotes the number of all distinct prime divisors of q . The casewhere q is a prime is due to Weil [4]. Zhang [5] proved that for any odd prime

A Prime Sensitive Hankel Determinant of Jacobi Symbol Enumerators

We show that the determinant of a Hankel matrix of odd dimension n whose entries are the enumerators of the Jacobi symbols which depend on the row and the column indices vanishes if and only if n is composite. If the dimension is a prime p, then the determinant evaluates to a polynomial of degree p − 1 which is the product of a power of p and the generating polynomial of the partial sums of Legendre symbols. The sign of the determinant is determined by the quadratic character of −1 modulo p. The proof of the evaluation makes use of elementary properties of Legendre symbols, quadratic Gauss sums, and orthogonality of trigonometric functions.

SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms

We propose a group-theoretical approach to the generalized oscillator algebra Aκ recently investigated in J. Phys. A: Math. Theor. 2010, 43, 115303. The case κ ≥ 0 corresponds to the noncompact group SU(1,1) (as for the harmonic oscillator and the Pöschl-Teller systems) while the case κ < 0 is described by the compact group SU(2) (as for the Morse system). We construct the phase operators and the corresponding temporally stable phase eigenstates for Aκ in this group-theoretical context. The SU(2) case is exploited for deriving families of mutually unbiased bases used in quantum information. Along this vein, we examine some characteristics of a quadratic discrete Fourier transform in connection with generalized quadratic Gauss sums and generalized Hadamard matrices.

Construction of equally entangled bases in arbitrary dimensions via quadratic Gauss sums and graph states

Recently [Karimipour and Memarzadeh, Phys. Rev. A 73, 012329 (2006)] studied the problem of finding a family of orthonormal bases in a bipartite space each of dimension $D$ with the following properties: (i) The family continuously interpolates between the product basis and the maximally entangled basis as some parameter $t$ is varied, and (ii) for a fixed $t$, all basis states have the same amount of entanglement. The authors derived a necessary condition and provided explicit solutions for $D \leq 5$ but the existence of a solution for arbitrary dimensions remained an open problem. We prove that such families exist in arbitrary dimensions by providing two simple solutions, one employing the properties of quadratic Gauss sums and the other using graph states. The latter can be generalized to multipartite equientangled bases with more than two parties.

Maximally Equiangular Frames and Gauss Sums

In a finite-dimensional complex Euclidean space, a maximally equiangular frame is a tight frame which has a number of elements equal to the square of the dimension of the space, and in which the inner products of distinct elements are of constant magnitude. Though the general question of their existence remains open, many examples of maximally equiangular frames have been constructed as finite Gabor systems. These constructions involve number theory, specifically Schaar’s identity, which provides a reciprocity formula for quadratic Gauss sums. To be precise, Zauner used Schaar’s identity to compute the spectrum of a chirp-Fourier operator, the eigenvectors of which he conjectured to be well-suited for the construction of maximally equiangular Gabor frames. We provide two new characterizations of such frames, both of which further confirm the relevance of the theory of Gauss sums to this area of frame theory. We also show how the unique time-frequency properties of a particular cyclic chirp function may be exploited to provide a new, short and elementary proof of Schaar’s identity.

Block-circulant matrices with circulant blocks, Weil sums, and mutually unbiased bases. II. The prime power case

In our previous paper [Combescure, M., “Circulant matrices, Gauss sums and the mutually unbiased bases. I. The prime number case,” Cubo A Mathematical Journal (unpublished)] we have shown that the theory of circulant matrices allows to recover the result that there exists p+1 mutually unbiased bases in dimension p, p being an arbitrary prime number. Two orthonormal bases B, B′ of Cd are said mutually unbiased if ∀b∊B, ∀b′∊B′ one has that |b⋅b′|=1/d (b⋅b′ Hermitian scalar product in Cd). In this paper we show that the theory of block-circulant matrices with circulant blocks allows to show very simply the known result that if d=pn (p a prime number and n any integer) there exists d+1 mutually unbiased bases in Cd. Our result relies heavily on an idea of Klimov et al. [“Geometrical approach to the discrete Wigner function,” J. Phys. A 39, 14471 (2006)]. As a subproduct we recover properties of quadratic Weil sums for p≥3, which generalizes the fact that in the prime case the quadratic Gauss sum properties follow from our results.

On an identity associated with Weil's estimate and its applications

In this paper we study the fourth mean value of generalized quadratic Gauss sums, and get an interesting identity associated with Weil's estimate, ∑ a = 2 p − 2 ∑ b = 1 p − 1 ( a 2 − b 2 p ) ( b 2 − 1 p ) = { 10 − 2 p , p ≡ 1 (mod 4) ; 2 p − 6 , p ≡ 3 (mod 4) . By virtue of the identity, higher-order mean values of generalized quadratic Gauss sums are obtained.

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.

SU2 Nonstandard Bases: Case of Mutually Unbiased Bases

This paper deals with bases in a finite-dimensional Hilbert space. Such a space can be realized as a subspace of the representation space of SU(2) corresponding to an irreducible representation of SU(2). The representation theory of SU(2) is reconsidered via the use of two truncated deformed oscillators. This leads to replace the familiar scheme {j^2, j_z} by a scheme {j^2, v(ra)}, where the two-parameter operator v(ra) is defined in the enveloping algebra of the Lie algebra su(2). The eigenvectors of the commuting set of operators {j^2, v(ra)} are adapted to a tower of chains SO(3) > C(2j+1), 2j integer, where C(2j+1) is the cyclic group of order 2j+1. In the case where 2j+1 is prime, the corresponding eigenvectors generate a complete set of mutually unbiased bases. Some useful relations on generalized quadratic Gauss sums are exposed in three appendices.

On the mean value of [formula omitted

Let χ be the Dirichlet character modulo q ⩾ 3 and L ( s , χ ) denote the corresponding Dirichlet L-function. The mean value of | L ′ L ( 1 , χ ) | 2 k is studied and a few asymptotic formulae are given. Hybrid mean value of L ′ L ( 1 , χ ) , general Kloosterman sums and general quadratic Gauss sums are considered.

Quadratic Gauss sums on matrices

Gauss sums over a finite field are generalized to ones over a matrix ring, related to numbers of solutions of diagonal matrix equations. A similar relation of quadratic Gauss sums of two types is obtained. We also consider quadratic diagonal matrix equations with scalar coefficients.

Quadratic Gauss sums on matrices

Gauss sums over a finite field are generalized to ones over a matrix ring, related to numbers of solutions of diagonal matrix equations. A similar relation of quadratic Gauss sums of two types is obtained. We also consider quadratic diagonal matrix equations with scalar coefficients.

The First Power Mean of the Inversion of L–Functions Weighted by Quadratic Gauss Sums

The main purpose of this paper is to use estimates for character sums and analytic methods to study the first power mean of the inversion of Dirichlet L–functions with the weight of general quadratic Gauss sums, and three asymptotic formulae are obtained.

On polynomial reciprocity law

The well-known law of quadratic reciprocity has over 150 proofs in print. We establish a relation between polynomial Jacobi symbols and resultants of polynomials over finite fields. Using this relation, we prove the polynomial reciprocity law and obtain a polynomial analogue of classical Burde's quartic reciprocity law. Under the use of our polynomial Poisson summation formula and the evaluation of polynomial exponential map, we get a reciprocity for the generalized polynomial quadratic Gauss sums.

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 hybrid mean value of the inversion of L-functions and general quadratic Gauss sums

AbstractThe main purpose of this paper is, using the estimates for character sums and the analytic method, to study the 2k-th power mean of the inversion of Dirichlet L-functions with the weight of general quadratic Gauss sums, and give two interesting asymptotic formulas.

Note on the quadratic Gauss sums

Let p be an odd prime and {χ(m) = (m/p)}, m = 0, 1, …, p − 1 be a finite arithmetic sequence with elements the values of a Dirichlet character χ modp which are defined in terms of the Legendre symbol (m/p), (m, p) = 1. We study the relation between the Gauss and the quadratic Gauss sums. It is shown that the quadratic Gauss sums G(k; p) are equal to the Gauss sums G(k, χ) that correspond to this particular Dirichlet character χ. Finally, using the above result, we prove that the quadratic Gauss sums G(k; p), k = 0, 1, …, p − 1are the eigenvalues of the circulant p × p matrix X with elements the terms of the sequence {χ(m)}.

Gauss sums and quantum mechanics

By adapting Feynman's sum over paths method to a quantum mechanical system whose phase space is a torus, a new proof of the Landsberg-Schaar identity for quadratic Gauss sums is given. In contrast to existing non-elementary proofs, which use infinite sums and a limiting process or contour integration, only finite sums are involved. The toroidal nature of the classical phase space leads to discrete position and momentum, and hence discrete time. The corresponding `path integrals' are finite sums whose normalisations are derived and which are shown to intertwine cyclicity and discreteness to give a finite version of Kelvin's method of images.

Identities on Quadratic Gauss Sums

Identities on Quadratic Gauss Sums

Identities on quadratic Gauss sums

Given a local field F F , each multiplicative character θ \theta of the split algebra F × F F \times F or of a separable quadratic extension of F F has an associated generalized Gauss sum γ θ F \gamma _\theta ^F . It is a complex valued function on the character group of F × × F {F^ \times } \times F , meromorphic in the first variable. We define a pairing between such Gauss sums and study its properties when F F is a nonarchimedean local field. This has important applications to the representation theory of G L ( 2 , F ) GL(2,F) and correspondences [ GL 3 ] [{\text {GL}}3] .