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Three topics in additive prime number theory. (English) Zbl 1230.11113

Jerison, David (ed.) et al., Current developments in mathematics, 2007. Somerville, MA: International Press (ISBN 978-1-57146-134-6/pbk). 1-41 (2009).
This survey describes three recent developments in prime number theory of interest to the author. The first topic discussed is the result of D. A. Goldston, J. Pintz and C. Y. Yıldırım [Ann. Math. (2) 170, No. 2, 819–862 (2009; Zbl 1207.11096)], that \[ \liminf_{n\rightarrow\infty}\frac{p_{n+1}-p_n}{\log p_n}=0. \] The survey discusses the historical background to this problem, describes the Selberg sieve in general, and motivates the strategy for its eventual resolution.
The second topic in the survey is the work of C. Mauduit and J. Rivat [Ann. Math. (2) 171, No. 3, 1591–1646 (2010; Zbl 1213.11025)], concerning the equidistribution in residue classes of the sum of digits function for primes. Indeed the survey gives a fairly detailed argument for the parity of the sum of digits function, in the case of primes represented in binary. This involves an exposition of Vaughan’s identity, and of a key \(L^{\infty}\) bound for the “exponential sum” (just a sum of powers of \(-1\) in this case) associated to Type-I sums for the problem. The corresponding Type-II sum is less elegant, but again a reasonably detailed treatment is given. Overall the reader who wants to study the Maudui–Rivat work will find this a very palatable introduction.
The final topic of the survey is the author’s work with T. Tao [Ann. Math. (2) 167, No. 2, 481–547 (2008; Zbl 1191.11025)] showing that the primes contain arbitrarily long arithmetic progressions, and its quantitative generalization [Ann. Math. (2) 171, No. 3, 1753–1850 (2010; Zbl 1242.11071)]. Again the historical background, in the form of a conjecture attributed to Dickson, is discussed. We are also introduced to some of the key features of the argument, taking one as far as the Gowers Inverse Conjecture. Naturally, in a survey of this type one can barely scratch the surface of such a deep body of theory, but it is to be hoped that this paper will prove a useful starting point for those wanting to learn more.
For the entire collection see [Zbl 1166.00310].

MSC:

11N05 Distribution of primes
11N36 Applications of sieve methods
11L20 Sums over primes
11B30 Arithmetic combinatorics; higher degree uniformity
11B25 Arithmetic progressions
11-02 Research exposition (monographs, survey articles) pertaining to number theory
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