On the random nature of (prime) number distribution
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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