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

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.

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.

@article{JTNB_2008__20_3_799_0,
     author = {Rivoal, Tanguy},
     title = {On the distribution of Hawkins{\textquoteright}~random {\textquotedblleft}primes{\textquotedblright}},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {799--809},
     publisher = {Universit\'e Bordeaux 1},
     volume = {20},
     number = {3},
     year = {2008},
     doi = {10.5802/jtnb.651},
     mrnumber = {2523318},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jtnb.651/}
}
Rivoal, Tanguy. On the distribution of Hawkins’ random “primes”. Journal de Théorie des Nombres de Bordeaux, Tome 20 (2008) no. 3, pp. 799-809. doi : 10.5802/jtnb.651. http://archive.numdam.org/articles/10.5802/jtnb.651/

[1] H. M. Bui, J. P. Keating, On twin primes associated with the Hawkins random sieve. J. Number Theory 119.2 (2006), 284–296. | MR 2250047 | Zbl 1135.11048

[2] H. Cramér, On the order of magnitude of the difference between consecutive prime numbers. Acta Arith. 2 (1936), 23–46. | Zbl 0015.19702

[3] L. Dickson, A new extension of Dirichlet’s theorem on prime numbers. Messenger of Math. 33 (1904), 155–161.

[4] G. H. Hardy, J. E. Littlewood, Some Problems of ’Partitio Numerorum.’ III. On the Expression of a Number as a Sum of Primes. Acta Math. 44 (1923), 1–70. | MR 1555183

[5] D. Hawkins, The random sieve. Math. Mag. 31 (1957/1958), 1–3. | MR 99321 | Zbl 0086.03502

[6] D. Hawkins, Random sieves. II. J. Number Theory 6 (1974), 192–200. | MR 345926 | Zbl 0287.10033

[7] C. C. Heyde, A loglog improvement to the Riemann hypothesis for the Hawkins random sieve. Ann. Probab. 6 (1978), no. 5, 870–875. | MR 503956 | Zbl 0414.60032

[8] C. C. Heyde, On asymptotic behavior for the Hawkins random sieve. Proc. AMS 56 (1976), 277–280. | MR 404177 | Zbl 0336.60030

[9] M. Loève, Probability theory, Third edition. D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1963. | MR 203748 | Zbl 0108.14202

[10] J. D. Lorch, A generalized Hawkins sieve and prime k-tuplets. Rocky Mountain J. Math. 37 (2007), no. 2, 533–550. | MR 2333384

[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. | MR 360490 | Zbl 0312.10034

[12] W. Neudecker, D. Williams, The ‘Riemann hypothesis’ for the Hawkins random sieve. Compositio Math. 29 (1974), 197–200. | Numdam | MR 399029 | Zbl 0312.10033

[13] G. Tenenbaum, Introduction à la théorie analytique et probabiliste des nombres, Deuxième édition. Cours Spécialisés, Société Mathématique de France, Paris, 1995. | MR 1366197 | Zbl 0880.11001

[14] M. C. Wunderlich, A probabilistic setting for prime number theory. Acta Arith. 26 (1974), 59–81. | MR 371834 | Zbl 0257.10033

[15] M. C. Wunderlich, The prime number theorem for random sequences. J. Number Theory 8 (1976), no. 4, 369–371. | MR 429799 | Zbl 0341.10036