Elliott-Halberstam conjecture
From Number
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-Vinogradov theorem | Elliott-Halberstam conjecture holds for  |
Proved assuming generalized Riemann hypothesis |
