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On the random nature of (prime) number distribution

Erika Alvarez·Jean Pestieau
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Abstract

Let pi(x) denote the number of primes smaller or equal to x. We compare sqrt{pi}(x) with sqrt{R}(x) and sqrt{li}(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 sqrt{pi}-sqrt{R} and indicate how li(x) can cross pi(x) for the first time.

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Erika Alvarez, Jean Pestieau. 2004-12-14. On the random nature of (prime) number distribution. https://arxiv.org/abs/math/0412288

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