Patterns of Primes in Arithmetic Progressions

János Pintz ()
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János Pintz: Hungarian Academy of Sciences, Alfréd Rényi Institute of Mathematics

A chapter in Number Theory – Diophantine Problems, Uniform Distribution and Applications, 2017, pp 369-379 from Springer

Abstract: Abstract After the proof of Zhang about the existence of infinitely many bounded gaps between consecutive primes the author showed the existence of a bounded d such that there are arbitrarily long arithmetic progressions of primes with the property that p ′ = p + d is the prime following p for each element of the progression. This was a common generalization of the results of Zhang and Green-Tao. In the present work it is shown that for every m we have a bounded m-tuple of primes such that this configuration (i.e. the integer translates of this m-tuple) appear as arbitrarily long arithmetic progressions in the sequence of all primes. In fact we show that this is true for a positive proportion of all m-tuples. This is a common generalization of the celebrated works of Green-Tao and Maynard/Tao.

Date: 2017
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-319-55357-3_19

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DOI: 10.1007/978-3-319-55357-3_19

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