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 |