2016/03/11 by Robert J. Lemke Oliver, K. Soundararajan, Kannan Soundararajan · 1 voice
Chemistry · Mathematics · #Analytic Number Theory Research #Biology #Chemistry #Combinatorics #Discrete mathematics #Distribution (mathematics) #Finite Group Theory Research #Genetics #Limits and Structures in Graph Theory #Mathematical analysis #Mathematics #Mod #Phenomenon #Physics #Quantum mechanics #Residue (chemistry) #Sequence (biology) #math.NT
paper · pdf · doi:10.1073/pnas.1605366113
arxiv published 2016/03/11 · arxiv updated 2016/05/30 · openalex created_date 2016/06/24 · openalex publication_date 2016/07/14 · openalex updated_date 2026/07/28
Significance Prime numbers play a central role in analytic number theory, and are well known to be very well distributed among the reduced residue classes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mrow> <mml:mtext>mod</mml:mtext> <mml:mo> </mml:mo> <mml:mi>q</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> . Surprisingly, the same does not appear to be true for sequences of consecutive primes, with different patterns occurring with wildly different frequencies. We formulate a precise conjecture, based on the Hardy−Littlewood conjectures, which explains this phenomenon. In particular, we predict that all patterns do occur their fair share of the time in the limit, but that there are secondary terms only very slowly tending to zero that create the observed biases.