Elliott-Halberstam conjecture

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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