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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).
According to an anecdote of uncertain reliability, [1] in primary school Carl Friedrich Gauss reinvented the formula (+) for summing the integers from 1 through , for the case =, by grouping the numbers from both ends of the sequence into pairs summing to 101 and multiplying by the number of pairs. Regardless of the truth of this story, Gauss ...
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.
For example, the quadratic character mod p described above takes the value 1 at each quadratic residue, and takes the value -1 at each quadratic non-residue. 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 ...
Most of the more elementary definitions of the sum of a divergent series are stable and linear, and any method that is both stable and linear cannot sum 1 + 2 + 3 + ⋯ to a finite value (see § Heuristics below). More advanced methods are required, such as zeta function regularization or Ramanujan summation.
The German edition includes all of Gauss's papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes. Footnotes referencing the Disquisitiones Arithmeticae are of the form "Gauss, DA, Art. n". Gauss, Carl Friedrich (1986).
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.
To put this in context of the next proof, the individual elements of the Gauss sum are in the cyclotomic field = but the above formula shows that the sum itself is a generator of the unique quadratic field contained in L. Again, since the quadratic Gauss sum is an algebraic integer, we can use modular arithmetic with it.