On the random nature of (prime) number distribution

2004/12/14 by Erika Alvarez, Alvarez, Erika, Jean Pestieau +1
Mathematics · #11A41 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11A41

paper · pdf · doi:10.48550/arxiv.math/0412288

arxiv created 2004/12/14 · arxiv updated 2009/12/01

Abstract

Let pi(x) denote the number of primes smaller or equal to x. We compare sqrtpi(x) with sqrtR(x) and sqrtli(x), where R(x) and li(x) are the Riemann function and the logarithmic integral, respectively. We show a regularity in the distribution of the natural numbers in terms of a phase related to sqrtpi-sqrtR and indicate how li(x) can cross pi(x) for the first time.

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