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Landau’s problems on primes

János Pintz (2009)

Journal de Théorie des Nombres de Bordeaux

At the 1912 Cambridge International Congress Landau listed four basic problems about primes. These problems were characterised in his speech as “unattackable at the present state of science”. The problems were the following :(1)Are there infinitely many primes of the form n 2 + 1 ?(2)The (Binary) Goldbach Conjecture, that every even number exceeding 2 can be written as the sum of two primes.(3)The Twin Prime Conjecture.(4)Does there exist always at least one prime between neighbouring squares?All these...

Levels of Distribution and the Affine Sieve

Alex Kontorovich (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

We discuss the notion of a “Level of Distribution” in two settings. The first deals with primes in progressions, and the role this plays in Yitang Zhang’s theorem on bounded gaps between primes. The second concerns the Affine Sieve and its applications.

Lois de répartition des diviseurs. IV

Gérald Tenenbaum (1979)

Annales de l'institut Fourier

Soit [ α , β ] un sous-intervalle de [ 0 , 1 ] ; on montre que la probabilité pour qu’un diviseur d’un entier n appartiennent à [ n α , n β ] possède une loi de distribution dont la mesure de répartition est atomique, à support inclus dans l’ensemble des nombres dyadiques.

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