On the distribution of Hawkins’ random “primes”

Tanguy Rivoal[1]

  • [1] Institut Fourier, CNRS UMR 5582, Université Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint-Martin d’Hères cedex, France.

Journal de Théorie des Nombres de Bordeaux (2008)

  • Volume: 20, Issue: 3, page 799-809
  • ISSN: 1246-7405

Abstract

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Hawkins introduced a probabilistic version of Erathosthenes’ sieve and studied the associated sequence of random “primes” ( p k ) k ≥ 1 . 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 k ≤ n such that p k + α - p k = α is almost surely equivalent to n / log ( n ) α , for a given fixed integer α ≥ 1 . 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 k ≤ n such that p k ∈ a ℕ + b is almost surely equivalent to n / a , for given fixed integers a ≥ 1 and 0 ≤ b ≤ a - 1 , which is an analogue of Dirichlet’s theorem.

How to cite

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Rivoal, Tanguy. "On the distribution of Hawkins’ random “primes”." Journal de Théorie des Nombres de Bordeaux 20.3 (2008): 799-809. <http://eudml.org/doc/10861>.

@article{Rivoal2008,
abstract = {Hawkins introduced a probabilistic version of Erathosthenes’ sieve and studied the associated sequence of random “primes” $(p_k)_\{k\ge 1\}$. 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 $k\le n$ such that $p_\{k+\alpha \}-p_k=\alpha $ is almost surely equivalent to $n/\log (n)^\{\alpha \}$, for a given fixed integer $\alpha \ge 1$. 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 $k\le n$ such that $p_k\in a\mathbb\{N\}+b$ is almost surely equivalent to $n/a$, for given fixed integers $a\ge 1$ and $0\le b\le a-1$, which is an analogue of Dirichlet’s theorem.},
affiliation = {Institut Fourier, CNRS UMR 5582, Université Grenoble 1, 100 rue des Maths, BP 74, 38402 Saint-Martin d’Hères cedex, France.},
author = {Rivoal, Tanguy},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {random sieve; Hawkins primes; distribution of primes},
language = {eng},
number = {3},
pages = {799-809},
publisher = {Université Bordeaux 1},
title = {On the distribution of Hawkins’ random “primes”},
url = {http://eudml.org/doc/10861},
volume = {20},
year = {2008},
}

TY - JOUR
AU - Rivoal, Tanguy
TI - On the distribution of Hawkins’ random “primes”
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2008
PB - Université Bordeaux 1
VL - 20
IS - 3
SP - 799
EP - 809
AB - Hawkins introduced a probabilistic version of Erathosthenes’ sieve and studied the associated sequence of random “primes” $(p_k)_{k\ge 1}$. 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 $k\le n$ such that $p_{k+\alpha }-p_k=\alpha $ is almost surely equivalent to $n/\log (n)^{\alpha }$, for a given fixed integer $\alpha \ge 1$. 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 $k\le n$ such that $p_k\in a\mathbb{N}+b$ is almost surely equivalent to $n/a$, for given fixed integers $a\ge 1$ and $0\le b\le a-1$, which is an analogue of Dirichlet’s theorem.
LA - eng
KW - random sieve; Hawkins primes; distribution of primes
UR - http://eudml.org/doc/10861
ER -

References

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  11. W. Neudecker, On twin “primes” and gaps between successive “primes” for the Hawkins random sieve. Math. Proc. Cambridge Philos. Soc. 77 (1975), 365–367. Zbl0312.10034MR360490
  12. W. Neudecker, D. Williams, The ‘Riemann hypothesis’ for the Hawkins random sieve. Compositio Math. 29 (1974), 197–200. Zbl0312.10033MR399029
  13. G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Deuxième édition. Cours Spécialisés, Société Mathématique de France, Paris, 1995. Zbl0880.11001MR1366197
  14. M. C. Wunderlich, A probabilistic setting for prime number theory. Acta Arith. 26 (1974), 59–81. Zbl0257.10033MR371834
  15. M. C. Wunderlich, The prime number theorem for random sequences. J. Number Theory 8 (1976), no. 4, 369–371. Zbl0341.10036MR429799

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