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.

1$\begingroup$ 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 $\endgroup$ – David White Sep 12 '11 at 20:48

$\begingroup$ I retagged historyoverview since the methods the OP seems particularly interested in the historical development (based on which methods he's asking for). $\endgroup$ – David White Sep 12 '11 at 20:50
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
For van der Corput's method and some developpments, including the BombieriIwaniec method, the book of Graham and Kolesnik "Van der Corput's method of exponential sums" is one of the best sources.

$\begingroup$ This is really nice little book, thanks :). $\endgroup$ – Ravinder May 20 '11 at 14:47
Ten Lectures on the interface of harmonic analysis and number theory by Hugh Montgomery covers these things (Chapters 2 and 3).
K. Chandrasekharan, Exponential sums in the development of number theory, pp. 726 in Proceedings of the International Conference on Number Theory (Moscow, 1971), Trudy Mat. Inst. Steklov 132 (1973).

$\begingroup$ I was able to see this in its entirety on google books here:books.google.com/… ! Also, the pages ran from 3 to 24... $\endgroup$ – Rob Harron Sep 13 '11 at 5:39

$\begingroup$ Ah, 324 is the page range in the AMS republication of that volume (1975). $\endgroup$ – KConrad Sep 13 '11 at 9:39
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.