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I'm interested in looking at the details of Gauss' method of determining the sign of the Gauss sum in his "Summatio Quarumdam Serierum Singularium", and I was wondering if anyone knew if there was an English or French translation available at all? I've tried searching on the internet and haven't found anything. Also, are there any modern presentations of the Gauss' method that you would recommend?

Thanks for your help.

edited Nov 17 2010 at 15:04

Andrey Rekalo
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asked Nov 17 2010 at 4:18

Lea M
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D M Bressoud, On the value of Gaussian sums, J Number Theory 13 (1981) 88–94, MR 82b:10046, "gives a new proof, closely paralleling that of Gauss...." – Gerry Myerson Nov 17 2010 at 4:58

Thanks-this looks like it will be really helpful – Lea M Nov 17 2010 at 5:28

Andrey Rekalo · Accepted Answer · 2010-12-04 07:17:53Z UTC

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"The determination of Gauss sums" by Berndt and Evans (Bull. Amer. Math. Soc., Vol. 5, Number 2 (1981), 107-129.) contains an exposition of the original proof due to Gauss. It also includes a short historical outline of various classical and modern proofs due to Dirichlet, Cauchy and others.

edited Dec 4 2010 at 7:17

answered Nov 17 2010 at 4:52

Andrey Rekalo
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	Thanks for linking to this article – Lea M Nov 17 2010 at 5:29
	You're welcome. – Andrey Rekalo Nov 17 2010 at 5:43

Franz Lemmermeyer · Answer 2 · 2010-12-04 19:05:04Z UTC

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An English translation of Baumgart's exposition of Gauss's fourth proof can be found here.

answered Dec 4 2010 at 19:05

Franz Lemmermeyer
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That looks great-thanks so much for providing the link. – Lea M Dec 7 2010 at 2:05

Carlo Beenakker · Answer 3 · 2010-12-03 19:52:57Z UTC

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there is a German translation: pages 463-495 in: http://www.amazon.co.uk/Untersuchungen-hohere-arithmetik-Chelsea-Publishing/dp/0821842137

answered Dec 3 2010 at 19:52

Carlo Beenakker
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(English + French)/2 = German? – Adam Hughes Dec 4 2010 at 1:04

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actually, i would say that English $-$ French $\to$ German. – S. Sra Dec 4 2010 at 1:52

Unfortunately I only speak a tiny bit of German (I've just taken a very basic introductory course), but thanks for pointing out that there is a German translation. – Lea M Dec 7 2010 at 2:07

English or French translation of Gauss' “Summatio Quarumdam Serierum Singularium”

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