Consecutive primes and Beatty sequences

2016/12/05 by William D. Banks, Banks, William D., Victor Z. Guo +1
Mathematics · #11B83 #11N05 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11B83 #msc:11N05

paper · pdf · doi:10.48550/arxiv.1612.01468

12 pages

arxiv created 2016/12/05 · arxiv updated 2016/12/06

Abstract

Fix irrational numbers α,α>1 of finite type and real numbers β,β≥ 0, and let B and B be the Beatty sequences B:=(\lfloorαm+β\rfloor)_{m≥ 1}\quadand B:=(\lfloorαm+β\rfloor)_{m≥ 1}. In this note, we study the distribution of pairs (p,p^\sharp) of consecutive primes for which p∈ B and p^\sharp∈ B. Under a strong (but widely accepted) form of the Hardy-Littlewood conjectures, we show that |\p≤ x:p∈ B and p^\sharp∈ B\|=(αα)^-1π(x)+O(x(log x)^-3/2+ε), where π(x) is the prime counting function.

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