Talk:Analytic number theory

Book recommendations?
Can someone recommend a book on analytic number theory?
 * This question was asked a long time ago, but for future reference: Two excellent books that come to mind are
 * Tom Apostol's Introduction to Analytic Number Theory (from Springer-Verlag)
 * Gerald Tenenbaum's Introduction to Analytic Number Theory (from Cambridge University Press)
 * I like both of these books a lot. Apostol's is excellent for the undergraduate, and Tenenbaum's for the graduate student. Cheers, Doctormatt 20:23, 20 April 2007 (UTC)


 * I recommend Richard Bellman's Analytic Number Theory, An Introduction. Addison Wesley 1980 ISBN 0-8053-0360-X. Lycurgus 19:45, 18 June 2007 (UTC)
 * Manin, Yu.I. and Panchishkin, A.A: Introduction to Modern Number Theory, 2nd Edition, Encyclopedia of Mathematical Sciences Volume 49, Springer, 2000
 * Bateman, Paul T. and Diamond, Harold G. : Analytic Number Theory, An Introductory Course, World Scientific — Preceding unsigned comment added by 41.136.223.113 (talk) 17:05, 22 March 2012 (UTC)

Error term given in the prime number theorem wrong
As has been pointed out already, this article is of very bad quality and needs rewriting. I did my bit by rewriting the section on the prime number theorem. It previously claimed that the error in the prime number theorem tended to zero as x went to infinity. This is the statement that the error term is o(1). This is definitely wrong! The Riemann Hypothesis itself is equivalent to an error term of $$O(x^{1/2+\varepsilon})$$ which is a lot weaker. —Preceding unsigned comment added by 123.120.160.228 (talk) 08:57, 25 December 2008 (UTC)

Rewrite
This article is in quite a bad state - I rewrote the lead section as a start, but I'm going to redo most of it in the next couple of days. I don't like the current selection of topics, so I'd like suggestions for how this sort of article should be organised. Early thoughts:

History (Dirichlet, Riemann, recent developments)

Significant results (PNT, Gauss circle problem, Goldston-Yildirim, zeta function results and consequences)

Main techniques (L-functions, circle method, sieve methods, probabilistic number theory)

In particular, I don't understand why Erdos has his own section - a great number theorist certainly, but probably not one of the three most significant contributors as this article suggests. In general, I feel that no sections should be people oriented - the article would be better focused if we concentrated on the mathematics, giving proper attribution and then people can follow up on those links if they want to see more what an individual has contributed.

Thoughts welcome! I'd like to know if anyone else is interested in the development of this article. Joth (talk) 12:39, 26 October 2010 (UTC)

Also, I'm not sure whether the Green-Tao theorem deserves to be called a major breakthrough in analytic number theory, rather than in additive combinatorics, say. Most of the 'traditional' analytic number theory is adapted from the Goldston-Yildirim work on small gaps between the primes, so perhaps this is a better result to mention. Joth (talk) 12:41, 26 October 2010 (UTC)

I agree with Joth on this one. Garald (talk) 12:49, 8 October 2011 (UTC)

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Jacobi, elliptic functions, and number theory
E.T.Bell: Men of Math chapter on Jacobi says that Jacobi made a hobby out of applying elliptic functions to number theory. E.g., Jacobi re-proved Lagrange's theorem and improved on it by saying in HOW MANY WAYS every positive integer was a sum of four squares. Is Bell right? How else did Jacobi apply elliptic functions to number theory?

Results?
'It is well known for its results on prime numbers (involving the Prime Number Theorem and Riemann zeta function) and additive number theory (such as the Goldbach conjecture and Waring's problem).' is a bit muddled. The Goldbach conjecture isn't a result! 31.52.254.181 (talk) 20:18, 29 March 2019 (UTC)