# 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

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topRivoal, 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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