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 sumsG(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−1 are the eigenvalues of the circulant p×p matrix X with elements the terms of the sequence {χ(m)}. 2000 Mathematics Subject Classification. Primary 11L05; Secondary 11T24, 11L10.


Introduction.
The notions of Gauss and quadratic Gauss sums play an important role in number theory with many applications [10].In particular, they are used as tools in the proofs of quadratic, cubic, and biquadratic reciprocity laws [5,7].
In this article, we study the relation between the quadratic Gauss sums and the Gauss sums related to a particular Dirichlet character defined in terms of the Legendre symbol and prove that the Gauss sums G(k, χ), k = 0, 1,...,p −1 which correspond to the Dirichlet character χ(m) = (m/p) are actually the quadratic Gauss sums G(k; p), (k, p) = 1.
More precisely, consider the finite arithmetic sequence {χ(m) = (m/p)} with elements the values of a Dirichlet character χ mod p which are defined in terms of the Legendre symbol (m/p), (m, p) = 1 and a circulant p × p matrix X with elements these values.If f (x) is a polynomial of degree p − 1 with coefficients the elements of the arithmetic sequence {χ(m)}, m = 0, 1,...,p − 1, then X = f (T ), where T is a suitable p × p circulant matrix, namely the rotational matrix; T is orthogonal, diagonalizable with eigenvalues the pth roots of unity.In addition, the matrices X, T have the same eigenvectors while if λ is an eigenvalue of T , then f (λ) is the eigenvalue of X that corresponds to the same eigenvector [3,12,13].
Finally, using the above results, we give an algebraic interpretation of the quadratic Gauss sums, which also leads to a different way of computing them, by proving that they are the eigenvalues of the circulant p × p matrix X.
Let C be the set of complex numbers, A an n × n matrix with entries in C and be a polynomial of degree n, where n is an integer greater than 1.
Proposition 2.1.If λ is an eigenvalue of the n × n matrix A that corresponds to the eigenvector v, then the n × n matrix as an eigenvalue that corresponds to the same eigenvector v.
is the characteristic polynomial of the matrix A with eigenvalues λ 1 ,...,λ n , then is the characteristic polynomial of the matrix f (A).
Proposition 2.3.If an n × n matrix A has n distinct eigenvalues, then so has the matrix f (A).Moreover, if the matrix A is diagonalized by an n×n matrix S, then f (A) is also diagonalized by S. Definition 2.4.Let m be an integer greater than 1, and suppose that (m, n) = 1.If x 2 ≡ n mod m is soluble, then we call n a quadratic residue mod m; otherwise we call n a quadratic nonresidue mod m.Definition 2.5 (Legendre's symbol).Let p be an odd prime, and suppose that p n.We let It is easy to see that if n ≡ n mod p and p n, then (n/p) = (n /p) which implies that the Legendre symbol is periodic with period p.
Definition 2.6.An n × n matrix whose rows come by cyclic permutations to the right of the terms of the arithmetic sequence {a i }, i = 0, 1,...,n− 1 is called a circulant matrix.
In case that the matrix A becomes The n × n matrix T , which is called the rotational matrix, is orthogonal, that is, T −1 = T , such that T n = I n and having as eigenvalues the nth roots of unity [3,12].Moreover, T is diagonalizable and if W is the n × n matrix whose columns are the eigenvectors of T , where w = e 2π i/n , then (2.11)

Gauss and quadratic Gauss sums.
In this section, we study the relation between the quadratic Gauss sums and the Gauss sums related to a particular Dirichlet character defined in terms of the Legendre symbol.
is called quadratic Gauss sum.
Proof.The number of solutions of the congruence and therefore which is the required result.

4.
The quadratic Gauss sums as eigenvalues of a suitable circulant matrix.In this section, we give an algebraic interpretation of the quadratic Gauss sums that correspond to a Dirichlet character χ mod p which is defined in terms of the Legendre symbol (m/p), (m, p) = 1.In fact, we prove that the quadratic Gauss sums G(k; p), (k, p) = 1, are the eigenvalues of the circulant p ×p matrix X with elements the values χ(m) = (m/p), (m, p) = 1.
Let now n = p be an odd prime, χ(m) = (m/p) be a Dirichlet character mod p that is defined in terms of the Legendre symbol (m/p), (m, p) = 1 and consider the circulant p × p matrix whose rows come by cyclic permutation to the right of the terms of the arithmetic sequence {χ(m)}, m = 0, 1,...,p − 1.

Proof. We can write T = (e p e 1 •••e
Thus, according to Proposition 2.1, the matrix X has the same eigenvectors with T , which are the row vectors where w = e 2π i/p , while its corresponding eigenvalues are . . . the quadratic Gauss sums. Notice that, equations (4.5) can be written in matrix notation as Furthermore, the p × p matrix whose columns are the eigenvectors of X, diagonalize X, that is, Remark 4.3.Since every Dirichlet character χ mod p is periodic mod p, it has a finite Fourier expansion [1,7], where the coefficients α p (k) are given by or equivalently If we consider now the Dirichlet character χ(m) = (m/p) which is defined in terms of the Legendre symbol (m/p), (m, p) = 1, then we deduce that the quadratic Gauss sum G(k; p) = G(k, χ), k = 0, 1,...,p − 1 is the Fourier transform of χ evaluated at k.

Conclusion.
We have shown that the quadratic Gauss sums G(k; p), (k, p) = 1 can be considered as the eigenvalues of a suitable circulant p × p matrix X with elements the terms of the arithmetic sequence {χ(m) = (m/p)}.This leads both to an algebraic characterization and also to a different way of computing the quadratic Gauss sums by calculating the roots of the characteristic polynomial that correspond to the matrix X.
Moreover, this new point of view for the quadratic Gauss sums gives, in many cases, an easier way to calculate them (to my best knowledge) instead of a direct computation, since one can find several methods for computing the eigenvalues of a matrix or the roots of a polynomial [2,6,9].

Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space.Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon.Basically, the phenomenon of Fermi acceleration (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles.His original model was then modified and considered under different approaches and using many versions.Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards.The topic includes both conservative and dissipative dynamics.Papers discussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned.Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/.Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System at http:// mts.hindawi.com/according to the following timetable: