Elliott-Halberstam conjecture

Statement

For $n$ a natural number and $a$ an integer relatively prime to $n$, consider the modular prime-counting function $x\mapsto \pi (x;n,a)$, which counts the number of prime numbers less than or equal to $x$ that are congruent to $a$ modulo $n$. Let $\pi (x)$ be the prime-counting function at $x$ and $\varphi (n)$ the Euler phi-function of $n$. Consider the expression:

$E(x;n):=\max _{(a,n)=1}\left|\pi (x;n,a)-{\frac {\pi (x)}{\varphi (n)}}\right|$

The intuition here is that the primes should be roulghy equally distributed between the various congruence classes modulo $n$, so the expected number of primes in each congruence class is $\pi (x)/\varphi (n)$. We are interested in the largest of the deviations from this expected value.

Then, the Elliott-Halberstam conjecture states that for every $\theta <1$ and $A>0$ there exists a constant $C>0$ such that

$\sum _{1\leq n\leq x^{\theta }}E(x;n)\leq {\frac {Cx}{\log ^{A}x}}$

Relation with other facts/conjectures

Name	Statement	Status
Bombieri-Vinogradov theorem	Elliott-Halberstam conjecture holds for $\theta <1/2$	Proved assuming generalized Riemann hypothesis