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Astérisque - Titles - 322 (2008) 225-240

Titles < 2008 < 322

Géométrie différentielle, physique mathématique, mathématiques et société (II)
Volume en l'honneur de Jean Pierre Bourguignon

Oussama Hijazi (éditeur)
Astérisque 322 (2008), xvi+256 pages
Buy the book
Presentation, Summary

Equidistribution and Primes
Peter Sarnak
Astérisque 322 (2008), 225-240

Résumé :
Équidistribution des nombres premiers
Nous commençons par l'examen de divers problèmes classiques concernant l'existence de nombres premiers ou de nombres avec peu de facteurs premiers, ainsi que quelques-uns des développéments clés vers la résolution de ces questions posées il y a bien longtemps. Ensuite, nous plaçons la théorie dans un contexte géométrique naturel et général d'actions sur le n-espace affine et nous indiquons ce qui peut être établi dans ce contexte. Les méthodes utilisées pour développer un crible combinatoire dans ce contexte impliquent les formes automorphes, les graphes d'expansion et, de manière inattendue, les combinatoires arithmétiques. Nous fournissons des applications aux problèmes classiques, tels que la divisibilité des aires des triangles pythagoriens et les courbures des circles dans un paquetage apollonien entier.

Mots-clefs : Nombres premiers, tamis, orbites affines, nombres de saturation, expanseurs et somme-produit

Abstract:
We begin by reviewing various classical problems concerning the existence of primes or numbers with few prime factors as well as some of the key developments towards resolving these long standing questions. Then we put the theory in a natural and general geometric context of actions on affine n-space and indicate what can be established there. The methods used to develop a combinational sieve in this context involve automorphic forms, expander graphs and unexpectedly arithmetic combinatorics. Applications to classical problems such as the divisibility of the areas of Pythagorean triangles and of the curvatures of the circles in an integral Apollonian packing, are given.

Keywords: Primes, sieves, affine orbits, saturation numbers, expanders and sum-product

Class. math. : 11Axx, 20Gxx


ISSN : 0303-1179
Publié avec le concours de : Centre National de la Recherche Scientifique

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