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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.

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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 '11 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 '11 at 20:50
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up vote 8 down vote accepted

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

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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).

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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.

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This is really nice little book, thanks :). –  Ravinder May 20 '11 at 14:47
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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).

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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 '11 at 5:39
    
Ah, 3-24 is the page range in the AMS republication of that volume (1975). –  KConrad Sep 13 '11 at 9:39
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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.

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