2003/04/29 by Doron Gepner, Gepner, Doron
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Number Theory (math.NT) #hep-th #math.NT
paper · pdf · doi:10.48550/arxiv.math/0304477
arxiv created 2003/05/18 · arxiv updated 2009/11/30
The probability of finding a prime multiplet, i.e., a sequence of primes p and p+ai, i=1... m, being all primes where p is some prime less than the integer n is naively 1/log(n)m+1. It is shown that, in reality, it is proportional to this probability by a constant factor which depends on ai and m but not on n, for large n. These constants are appellated as PDF (prime distribution factors). Moreover, it is argued that the PDF depend on the ai in a "week" way, only on the prime factors of the differences ai-aj and not on their exponents. For example p and p+2s will have the exact same probability for all integer s>0. The exact formulae for the PDF ratios are given. Moreover, the actual 'basic' PDF's are calculated exactly and are shown to be bigger than 1, which indicates that primes 'repel' each other. An exact asymptotic formula for the number of basic multiplets is given.