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Prime numbers with Beatty sequences

William D. Banks, Igor E. Shparlinski (2009)

Colloquium Mathematicae

A study of certain Hamiltonian systems has led Y. Long to conjecture the existence of infinitely many primes which are not of the form p = 2⌊αn⌋ + 1, where 1 < α < 2 is a fixed irrational number. An argument of P. Ribenboim coupled with classical results about the distribution of fractional parts of irrational multiples of primes in an arithmetic progression immediately implies that this conjecture holds in a much more precise asymptotic form. Motivated by this observation, we give an asymptotic...

Progressions arithmétiques dans les nombres premiers

Bernard Host (2004/2005)

Séminaire Bourbaki

Récemment, B. Green et T. Tao ont montré que : l’ensemble des nombres premiers contient des progressions arithmétiques de toutes longueurs répondant ainsi à une question ancienne à la formulation particulièrement simple. La démonstration n’utilise aucune des méthodes “transcendantes” ni aucun des grands théorèmes de la théorie analytique des nombres. Elle est écrite dans un esprit proche de celui de la théorie ergodique, en particulier de celui de la preuve par Furstenberg du théorème de Szemerédi,...

Restriction theory of the Selberg sieve, with applications

Ben Green, Terence Tao (2006)

Journal de Théorie des Nombres de Bordeaux

The Selberg sieve provides majorants for certain arithmetic sequences, such as the primes and the twin primes. We prove an L 2 – L p restriction theorem for majorants of this type. An immediate application is to the estimation of exponential sums over prime k -tuples. Let a 1 , ⋯ , a k and b 1 , ⋯ , b k be positive integers. Write h ( θ ) : = ∑ n ∈ X e ( n θ ) , where X is the set of all n ≤ N such that the numbers a 1 n + b 1 , ⋯ , a k n + b k are all prime. We obtain upper bounds for ∥ h ∥ L p ( 𝕋 ) , p &gt; 2 , which are (conditionally on the Hardy-Littlewood prime tuple conjecture) of the correct order...

Currently displaying 101 – 120 of 159