On the distribution of Hawkins’ random “primes”
Journal de Théorie des Nombres de Bordeaux, Volume 20 (2008) no. 3, pp. 799-809.

Hawkins introduced a probabilistic version of Erathosthenes’ sieve and studied the associated sequence of random “primes” (p k ) k1 . Using various probabilistic techniques, many authors have obtained sharp results concerning these random “primes”, which are often in agreement with certain classical theorems or conjectures for prime numbers. In this paper, we prove that the number of integers kn such that p k+α -p k =α is almost surely equivalent to n/log(n) α , for a given fixed integer α1. This is a particular case of a recent result of Bui and Keating (differently formulated) but our method is different and enables us to provide an error term. We also prove that the number of integers kn such that p k a+b is almost surely equivalent to n/a, for given fixed integers a1 and 0ba-1, which is an analogue of Dirichlet’s theorem.

Hawkins a défini une version probabiliste du crible d’Ératosthène et étudié la suite des nombres “premiers” aléatoires (p k ) k1 ainsi créés. Au moyen de diverses techniques probabilistes, de nombreux auteurs ont ensuite obtenu des résultats très fins sur ces “premiers”, souvent en accord avec des théorèmes ou conjectures classiques sur les nombres premiers usuels. Dans ce papier, on prouve que le nombre d’entiers kn tel que p k+α -p k =α est presque sûrement équivalent à n/log(n) α , pour tout entier α1 fixé. C’est un cas particulier d’un travail récent de Bui and Keating (exprimé autrement) mais notre méthode est différente et fournit un terme d’erreur. On montre également que le nombre d’entiers kn tel que p k a+b est presque sûrement équivalent à n/a, pour tous entiers a1 et 0ba-1 fixés, ce qui peut être vu comme un analogue du théorème de Dirichlet.

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DOI: 10.5802/jtnb.651
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Tanguy Rivoal. On the distribution of Hawkins’ random “primes”. Journal de Théorie des Nombres de Bordeaux, Volume 20 (2008) no. 3, pp. 799-809. doi : 10.5802/jtnb.651. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.651/

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