O Crivo de Brun para primos g?meos.

Abstract

Programa de P?s-Gradua??o em Matem?tica em Rede Nacional. Departamento de Matem?tica, Instituto de Ci?ncias Exatas e Biol?gicas, Universidade Federal de Ouro Preto.Um n?mero primo p ? dito primo g?meo se p+2 tamb?m ? primo. Conjectura-se que existem infinitos primos g?meos. O objetivo dessa disserta??o ? mostrar que a soma dos inversos dos primos g?meos converge, enquanto a soma dos inversos de todos os primos diverge (tamb?m vamos provar isso usando a fun??o zeta de Riemann). Tal fato pode implicar duas coisas: ou existem finitos primos g?meos, ou os primos g?meos s?o infinitos por?m muito escassos na reta real. A t?cnica utilizada para demonstrar esse resultado ? o crivo de Brun, que permite obter uma cota superior para o n?mero de primos g?meos at? x. Para obter tais cotas, ? necess?rio apresentar diversos resultados anteriores, como o princ?pio da inclus?oexclus?o, as fun??es multiplicativas (em particular, a fun??o de M?bius), as duas primeiras f?rmulas de Mertens e o Teorema de Chebyshev. Vamos apresentar tamb?m uma caracteriza??o dos primos g?meos devida a Clement. A cota superior obtida implica diretamente o principal resultado dessa disserta??o: a soma dos inversos dos primos g?meos converge.A prime number p is said to be a twin prime if p+2 is also a prime. It?s conjectured that there exist infinitely many twin primes. The goal of this master thesis is to show that the sum of the inverses of the twin primes converges, although the sum of the inverses of all the primes diverges (we will also prove the latter using the Riemann zeta function). Such fact can imply two things: either there are finitely many twin primes, or there are infinitely many, however sparse on the real line. The technique used to prove this result is the Brun?s sieve, which allows us to obtain an upper bound for the number of twin primes up to x. In order to obtain such bounds, it?s required to present several preliminary results, such as inclusionexclusion principle, the multiplicative functions (in particular, the M?bius function), the first two Merten?s formula and the Chebyshev Theorem. We will also present a characterization of twin primes due to Clement. The obtained upper bound directly implies the main result of this thesis: the sum of inverses of twin primes converges

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This paper was published in REPOSITORIO INSTITUCIONAL DA UFOP.

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