ON THE DISTRIBUTION OF HAWKINS' RANDOM “PRIMES”

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Niveau: Supérieur, Doctorat, Bac+8
ON THE DISTRIBUTION OF HAWKINS' RANDOM “PRIMES” TANGUY RIVOAL Dedicated to Henri Cohen on the occasion of his 60th birthday Abstract. Hawkins introduced a probabilistic version of Era- thostenes' sieve and studied the associated sequence of random “primes” (pk)k≥1. Using various probabilistic techniques, many authors have obtained sharp results concerning these random “pri- mes”, 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 pk+?? pk = ? is almost surely equivalent to n/ log(n)?, for a given fixed integer ? ≥ 1. This is a particular case of a recent result of Bui and Keating (differently expressed) 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 pk ? aN+ 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. Resume. Hawkins a defini une version probabiliste du crible d'Era- tosthene et etudie la suite des nombres “premiers” aleatoires (pk)k≥1 ainsi crees. Au moyen de diverses techniques probabilistes, de nom- breaux auteurs ont ensuite obtenu des resultats tres fins sur ces “premiers”, souvent accord avec des theoremes ou conjectures clas- siques sur les nombres premiers usuels.

  • version probabiliste du crible d'era- tosthene

  • integers

  • prime numbers

  • prime

  • still conjectures

  • analogue du theoreme de dirichlet

  • erathostenes' sieve


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ON THE DISTRIBUTION OF HAWKINS’ RANDOM “PRIMES”
TANGUY RIVOAL
th Dedicated to Henri Cohen on the occasion of his 60 birthday
Abstract.Hawkins introduced a probabilistic version of Era-thostenes’ sieve and studied the associated sequence of random “primes” (pk)k1. Using various probabilistic techniques, many authors have obtained sharp results concerning these random “pri-mes”, which are often in agreement with certain classical theorems or conjectures for prime numbers. In this paper, we prove that the number of integersknsuch thatpk+αpk=αis almost surely α equivalent ton/log(n) , for a given fixed integerαis1. This a particular case of a recent result of Bui and Keating (differently expressed) but our method is different and enables us to provide an error term. We also prove that the number of integerskn such thatpkaN+bis almost surely equivalent ton/a, for given fixed integersa1 and 0ba1, which is an analogue of Dirichlet’s theorem.
´ Re´sume´.HsndawaikbledEra-aborilibdetsircun´eneiursvenpio tosthe`neet´etudi´elasuitedesnombrespremiersal´eatoires(pk)k1 ainsicr´ee´s.Aumoyendediversestechniquesprobabilistes,denom-breauxauteursontensuiteobtenudesre´sultatstre`snssurces premiers,souventaccordavecdesthe´ore`mesouconjecturesclas-siques sur les nombres premiers usuels. Dans ce papier, on prouve que le nombre d’entierskntel quepk+αpk=αest presque α suˆremente´quivalent`an/log(npour tout entier) , α.C´ex1ste uncasparticulierduntravailre´centdeBuiandKeating(exprime´ autrement)maisnotrem´ethodeestdie´renteetfournitunterme derreur.Onmontre´egalementquelenombredentierskntel quepkaN+blaviuqe´a`tnepresquesˆurementetsn/a, pour tous entiersa1 et 0baquipns,cememterueevuutcoˆxe´1 analogueduth´eor`emedeDirichlet.
1.Introduction The simplest method for determining a not too large list of prime numbers is Erathostenes’ sieve. Legendre found an analytical for-mula for this sieve which can theoretically be used to compute any
Date: October 28, 2008.
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