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The case originally considered by Carl Friedrich Gauss was the quadratic Gauss sum, for R the field of residues modulo a prime number p, and χ the Legendre symbol.In this case Gauss proved that G(χ) = p 1 ⁄ 2 or ip 1 ⁄ 2 for p congruent to 1 or 3 modulo 4 respectively (the quadratic Gauss sum can also be evaluated by Fourier analysis as well as by contour integration).
In number theory, quadratic Gauss sums are certain finite sums of roots of unity. A quadratic Gauss sum can be interpreted as a linear combination of the values of the complex exponential function with coefficients given by a quadratic character; for a general character, one obtains a more general Gauss sum.
In mathematics, Gauss congruence is a property held by certain sequences of integers, including the Lucas numbers and the divisor sum sequence. Sequences satisfying this property are also known as Dold sequences, Fermat sequences, Newton sequences, and realizable sequences. [ 1 ]
The Hasse–Davenport relations, introduced by Davenport and Hasse , are two related identities for Gauss sums, one called the Hasse–Davenport lifting relation, and the other called the Hasse–Davenport product relation. The Hasse–Davenport lifting relation is an equality in number theory relating Gauss sums over different fields.
In mathematics, the Gross–Koblitz formula, introduced by Gross and Koblitz expresses a Gauss sum using a product of values of the p-adic gamma function. It is an analog of the Chowla–Selberg formula for the usual gamma function. It implies the Hasse–Davenport relation and generalizes the Stickelberger theorem.
The Gauss sum (,) can thus be written as a linear combination of Gaussian periods (with coefficients χ(a)); the converse is also true, as a consequence of the orthogonality relations for the group (Z/nZ) ×. In other words, the Gaussian periods and Gauss sums are each other's Fourier transforms.
VNU University of Science (VNU-HUS; Vietnamese: Trường Đại học Khoa học Tự nhiên, Đại học Quốc gia Hà Nội) or Hanoi University of Science, is a member of Vietnam National University, Hanoi. Founded in 1906 as Indochina University, the university has changed its name three times: School of Basic Science (1951), University ...
In mathematics, addition and subtraction logarithms or Gaussian logarithms can be utilized to find the logarithms of the sum and difference of a pair of values whose logarithms are known, without knowing the values themselves. [1] Their mathematical foundations trace back to Zecchini Leonelli [2] [3] and Carl Friedrich Gauss [4] [1] [5] in the ...