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What are the good books, online lecture notes or starting material on exponentials sums with applications in number theory for a beginner, apart from N. M. Korobov's book? The book or notes should cover methods of Weyl, van der Corput and Vinogradov, with some details.

edited Sep 12 2011 at 20:50

David White

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First, I think this should be CW since it's asking for a list of good books. Second, if you're no longer a beginner, having read the below references, then I recommend Serre's A Course in Arithmetic for some of the higher level applications of exponential sums. Third, there seem to be some nice references for beginners at this link: mathworld.wolfram.com/WeylSum.html – David White Sep 12 2011 at 20:48

I retagged history-overview since the methods the OP seems particularly interested in the historical development (based on which methods he's asking for). – David White Sep 12 2011 at 20:50

Felipe Voloch · Accepted Answer · 2011-05-19 14:23:32Z UTC

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Shparlinski has a nice set of lecture notes, aimed at beginners, with a view towards applications: http://www2.ims.nus.edu.sg/Programs/coding/files/ishpar.ps

answered May 19 2011 at 14:23

Felipe Voloch

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KConrad · Answer 2 · 2011-09-13 04:05:36Z UTC

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K. Chandrasekharan, Exponential sums in the development of number theory, pp. 7-26 in Proceedings of the International Conference on Number Theory (Moscow, 1971), Trudy Mat. Inst. Steklov 132 (1973).

answered Sep 13 2011 at 4:05

KConrad

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	I was able to see this in its entirety on google books here:books.google.com/… ! Also, the pages ran from 3 to 24... – Rob Harron Sep 13 2011 at 5:39
	Ah, 3-24 is the page range in the AMS republication of that volume (1975). – KConrad Sep 13 2011 at 9:39

Denis Chaperon de Lauzières · Answer 3 · 2011-05-19 17:25:08Z UTC

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For van der Corput's method and some developpments, including the Bombieri-Iwaniec method, the book of Graham and Kolesnik "Van der Corput's method of exponential sums" is one of the best sources.

answered May 19 2011 at 17:25

Denis Chaperon de Lauzières

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This is really nice little book, thanks :). – Ravinder May 20 2011 at 14:47

Helge · Answer 4 · 2011-05-19 14:45:46Z UTC

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Ten Lectures on the interface of harmonic analysis and number theory by Hugh Montgomery covers these things (Chapters 2 and 3).

answered May 19 2011 at 14:45

Helge

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arithboy · Answer 5 · 2011-05-19 13:46:15Z UTC

Maybe you will find something useful in the book of Iwaniec & Kowalski entitled " Analytic Number Theory". Morever, I think Huxley's "Area, Lattice Points and Exponential Sums" is worthwhile to read, which focuses a new method deveploed by Bombieri, Iwaniec, Huxley himself and many other followers.

Exponential sums for beginner.

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