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The quadratic Gauss sum redux. (English) Zbl 1378.11011

Gauss sums, and quadratic Gauss sums in particular, are mysterious objects. Mysterious not because they we do not understand them properly but because they show up in so many different connections.
The basic property of the quadratic Gauss sum \(g = \sum (\frac ap) \zeta^a\), where \(p\) is an odd prime, \((\frac ap)\) the Legendre symbol and \(\zeta\) a primitive \(p\)-th root of unity, is the equation \(g^2 = (\frac{-1}p) p\). Gauss has shown that this relation implies the quadratic reciprocity law, and the author sketches various approaches for proving this identity that range from algebraic manipulations to functional equations of L-series or the Weil conjectures. As Gauss discovered the hard way, it is much more difficult to determine the sign of the Gauss sum \(g\) (which makes sense only once \(\zeta = \exp(\frac{2\pi i}p)\) has been fixed analytically); Schur’s beautiful determination of \(g\) is sketched briefly.
Readers who, like the reviewer, regret that this article is only a few pages long, will find a lot more material in the article by B. C. Berndt and R. J. Evans [Bull. Am. Math. Soc., New Ser. 5, 107–129 (1981; Zbl 0471.10028)] and the book by B. C. Berndt et al. [Gauss and Jacobi sums. New York, NY: John Wiley & Sons (1998; Zbl 0906.11001)] or, if they can manage to find a copy, in the thesis by M. Salvadori [Esposizione della teoria delle somme di Gauss. Diss. (Freiburg, Schweiz). Pisa, Tipografia fratelli Nistri. 116 p. (1904; JFM 35.0210.01); Esposizione della teoria delle somme di Gauss e di alcuni teoremi di Eisenstein (Italian). Pisa: Nistri (1904; JFM 35.0977.02)].

MSC:

11A15 Power residues, reciprocity
11L05 Gauss and Kloosterman sums; generalizations
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