The Elliott-Halberstam conjecture is a conjecture about the average distribution of prime numbers in arithmetic progressions. Let
 |
where 
is the sum of the Mangoldt function over integers
with 
,
and 
is the totient function. The conjecture states
that for every 
and 
,
 |
where the implied constant may depend on 
and 
(Elliott and Halberstam 1970).
The Bombieri-Vinogradov theorem proves the corresponding assertion for every 
, whereas the Elliott-Halberstam conjecture asserts
it for every 
. The generalized Elliott-Halberstam conjecture is
stronger. It extends analogous distribution estimates to suitable convolutions
of arithmetic functions.
The conjectures have consequences for bounded gaps between primes. If 
is the 
th prime, write 
. Elliott-Halberstam implies
(Maynard 2015) and 
(Polymath 2014), the latter improving Maynard's
earlier bound 
. The generalized conjecture implies 
and 
(Polymath 2014). Both statements are conditional.
Neither gives 
, so neither proves the twin
prime conjecture.
