Elliott-Halberstam conjecture

From Number
Revision as of 03:09, 9 February 2010 by Vipul (talk | contribs) (Created page with '==Statement== For <math>n</math> a natural number and <math>a</math> an integer relatively prime to <math>m</math>, consider the [[fact about::modular prime-counting functio…')
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)

Statement

For a natural number and an integer relatively prime to , consider the modular prime-counting function , which counts the number of prime numbers less than or equal to that are congruent to modulo . Let be the prime-counting function at and the Euler phi-function of . Consider the expression:

The intuition here is that the primes should be roulghy equally distributed between the various congruence classes modulo , so the expected number of primes in each congruence class is . We are interested in the largest of the deviations from this expected value.

Then, the Elliott-Halberstam conjecture states that for every and there exists a constant such that

Relation with other facts/conjectures

Name Statement Status
Bombieri-Vingradov theorem Elliott-Halberstam conjecture holds for Proved assuming generalized Riemann hypothesis