Granville, Andrew (ed.) et al., Equidistribution in number theory, an introduction. Proceedings of the NATO Advanced Study Institute on equidistribution in number theory, Montréal, Canada, July 11–22, 2005. Dordrecht: Springer (ISBN 978-1-4020-5403-7/pbk; 978-1-4020-5402-0/hbk). NATO Science Series II: Mathematics, Physics and Chemistry 237, 59-83 (2007).
As the title alludes to, the author gives a survey on the state-of-the-art literature on the asymptotic features of prime numbers.
In fact, the paper is divided into three sections:
1. The Cramèr model and gaps between consecutive primes;
2. The distribution of primes in longer intervals;
3. Maier’s method and an “uncertainty principle”.
In the first section the Cramèr model is exposed, with its limitations; which are of an arithmetical nature, as shown (conjecturally) by Hardy and Littlewood in their milestone paper named Partitio Numerorum, III of 1922 (curiously enough, Cramèr proposed his model in 1936, actually!).
Using the Hardy-Littlewood conjecture (see the paper under review or, better, Conjecture B of the 1922 article), though, Gallagher in 1976 corrected mathematicians’ expectations for Cramèr’s model to be true, to a milder statement: Cramèr model is true on average (not individually, see upside).
Then, the first section concludes with a glimpse to the recent result of Goldston-Yildirim-Pintz (a real milestone!) on small gaps between consecutive primes.
On the other side (of longer intervals), section 2 deals with moments of primes in short intervals (yes, short, but however of length not smaller than logarithm powers of the endpoints); showing the connections with arguments of section 1 and, also, with the zeros of the Riemann function.
A finer study, then, passes from this to Dirichlet’s functions, because (and, now, we enter section 3) Maier discovered a link between primes in short intervals and their distribution in arithmetic progressions (of very small modulus, say powers of the logarithm of our variable, again linked to endpoints of the short interval), which is governed (as unveiled by Dirichlet himself) by the zeros of these -functions.
Maier’s “Matrix method ” marks (in 1985) another milestone for our knowledge about primes.
In essence, it gives incompatibility (an “uncertainty principle”) between two “primes regularities”:
good distribution in “short” intervals and, on the other side, into arithmetic progressions (of “small” moduli).
This incompatibility is a general phenomenon (not only belonging to primes), as discovered by Granville and others: see the remarks concluding section three.