Around an idea of Bombieri on the Selberg sieve

Description

Around an idea of Bombieri on the Selberg sieve V. Devinck June 28, 2009

• viggo brun


• prime twins


• integer r0


• over all


• without small


• sieve method


• function ?


• r0 prime factors

Sujets

• Viggo Brun
• Twin prime
• Sieve method

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 43
• Langue: English

Extrait

Around

an

idea

of

Bombieri sieve

V. Devinck

June

28,

2009

on

the

Selberg

2

Introduction

In these notes, we are interested in the number of prime factors of two se- quences of integers and more precisely, we aim at giving an upper bound of this number by using the weighted sieve. In the first example, we will con- sider the integersp+ 2 whenpis a prime number; as of today, the problem of prime twins (”there is infinitely many primespsuch thatp+ 2 is prime”) remains a mystery. Nevertheless, we have some ideas about the quantity of prime twins: they are, in some sense, rare: Viggo Brun (1882-1978) proved that the sum of the reciprocals of twin primes is convergent or finite which is in contrast with the fact that the sum of the reciprocals over all prime numbers is divergent. Let{a n } n≥1  look at the set Webe a sequence of (positive) integers. E(x) :={n≤x|ω(a n )≤r 0 }(xpositive real number) of integersa n such that a n has at mostr 0 prime factors for some positive integerr 0 (andn≤x). The general problem is to determine an admissible value ofr 0 such that #{n≥2|ω(a n )≤r 0 }= +∞. The best result in this direction for the sequence{p+ 2} p≥2  1973) withis due to J.-R. Chen (Sci. Sinica,r 0  For= 2. the sequence{n(n+ 2)} n≥1 (which is our second application of the weighted sieve)thebestresultisattributedtoRe´nyiwiththeintegerr 0  We will= 5. further prove a lower bound of the cardinality ofE(x); the boundedness of the weights in the weighted sieve will allow us to obtain such a result. Moreover, we refine our results by showing that we can assumep+ 2 (respectively n(n+ 2))have no prime factor less than a small power of to p(respectively n), which we explicitly determine.

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