# Elliott-Halberstam conjecture

## 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