Dense clusters of primes in subsets

2014/05/11 by James Maynard, Maynard, James · 7 citations
Mathematics · #Analytic Number Theory Research #Limits and Structures in Graph Theory #Finite Group Theory Research

paper · pdf · doi:10.1112/s0010437x16007296

Abstract

We prove a generalization of the author’s work to show that any subset of the primes which is ‘well distributed’ in arithmetic progressions contains many primes which are close together. Moreover, our bounds hold with some uniformity in the parameters. As applications, we show there are infinitely many intervals of length (log x)^\itε containing ≫ _\itεlog log x primes, and show lower bounds of the correct order of magnitude for the number of strings of m congruent primes with p_n+m-p_n\leqslant \itεlog x .

Cited by

• Short proofs in combinatorics and number theory
• Small gaps between the Piatetski-Shapiro primes
• d-Translated Unit Sensitive Primes
• Prime-free discs in imaginary quadratic fields
• Arithmetic progressions of primes in short intervals beyond the 17/30 barrier
• Primes in intervals of bounded length
• Bounded gaps between primes in special sequences

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• Arithmetic progressions of primes in short intervals beyond the 17/30 barrier
• Primes in Tuples I
• Short intervals containing a prescribed number of primes
• Clusters of primes with square-free translates
• On the prime distribution