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Bounded gaps between primes in number fields and function fields. (English) Zbl 1321.11098

We quote the authors’ abstract: “The Hardy-Littlewood prime \(k\)-tuples conjecture has long been thought to be completely unapproachable with current methods. While this sadly remains true, startling breakthroughs of Zhang, Maynard, and Tao have nevertheless made significant progress toward this problem. In this work, we extend the Maynard-Tao method to both number fields and the function field \({\mathbb F}_q(t)\).”
In fact, the authors quote the recent Maynard-Tao theorem; then they give the same result in the two new environments (instead of the rational integers), namely, the number fields \(K\) (in which the rôle of primes is taken by elements of the ring of integers of \(K\) generating principal prime ideals) and the function fields \({\mathbb F}_q(t)\) (now we take the monic irreducible polynomials in \({\mathbb F}_q[t]\) as primes).
The proofs follow along the lines of Maynard-Tao theorem’s proof (but now the positive level of distribution of primes in arithmetic progressions has more technical analogs, the two counterparts given in Theorem 2.1 and 2.2), of course adapting to the two new different environments: the reader has a “dictionary”, for the translations in section 2.1.
Last but not least, in Section 3 they prove two other complementary results, namely Corollary 1.2 (for number fields) and Theorem 1.4 (for \({\mathbb F}_q(t)\) instead) which have their own interest.

MSC:

11N05 Distribution of primes
11N36 Applications of sieve methods
11T06 Polynomials over finite fields
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References:

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