Patterns of primes in arithmetic progressions

Pintz, János (2017) Patterns of primes in arithmetic progressions. In: Number Theory. Springer, Berlin; Heidelberg; New York, pp. 369-379. ISBN 978-3-319-55356-6

Preview

Text 1004.1078v1.pdf Download (100kB) | Preview

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. Dedicated to the 60th birthday of Robert F. Tichy

Actions (login required)

Edit Item