ON THE DISTRIBUTION OF HAWKINS' RANDOM “PRIMES”

Henri Cohen

Description

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

Sujets

• Integer
• Prime number

Informations

• Publié par: mijec
• Nombre de lectures: 75
• Langue: English

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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” (p k ) 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 integersk≤nsuch thatp k+α −p k =αis almost surely α equivalent ton/log(n) , for a given fixed integerα≥is1. 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 integersk≤n such thatp k ∈aN+bis almost surely equivalent ton/a, for given fixed integersa≥1 and 0≤b≤a−1, which is an analogue of Dirichlet’s theorem.

´ Re ´ sume ´ .Hsndawaikbled’Era-aborilibdetsircufin´eneiursvenpio tosthe`neet´etudi´elasuitedesnombres“premiers”al´eatoires(p k ) k≥1 ainsicr´ee´s.Aumoyendediversestechniquesprobabilistes,denom- breauxauteursontensuiteobtenudesre´sultatstre`sfinssurces “premiers”,souventaccordavecdesthe´ore`mesouconjecturesclas- siques sur les nombres premiers usuels. Dans ce papier, on prouve que le nombre d’entiersk≤ntel quep k+α −p k =αest presque α suˆremente´quivalent`an/log(npour tout entier) , α≥.C´efix1st’e uncasparticulierd’untravailre´centdeBuiandKeating(exprime´ autrement)maisnotrem´ethodeestdiffe´renteetfournitunterme d’erreur.Onmontre´egalementquelenombred’entiersk≤ntel quep k ∈aN+blaviuqe´a`tnepresquesˆurementetsn/a, pour tous entiersa≥1 et 0≤b≤a−quipns,cememterueevuutcoˆxfie´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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