The Elliott–Halberstam Conjecture

Conjecture (level 1): for every θ < 1 and every A > 0, \sum_{q ≤ x^θ} \(max_{(a,q)=1}\) |π(x;q,a) − li(x)/φ(q)| ≪_A x/(log x)^A. Bombieri–Vinogradov proves θ = 1/2.

The engine behind bounded gaps between primes: Zhang's 2013 breakthrough needed distribution past 1/2, and Maynard–Tao's method converts any available level into gap bounds — Polymath8b reached gaps ≤ 246 using θ = 0.525. Bombieri–Friedlander–Iwaniec pushed special cases beyond 1/2; full level 1 stays open, with the generalised-EH variants now a cottage industry.