The Bombieri-Vinogradov theorem, also called Bombieri's theorem, describes the average distribution of prime numbers in arithmetic progressions. Define
 |
(1)
|
where
 |
(2)
|
(Davenport 1980, p. 121), 
is the Mangoldt function,
and 
is the totient function. Now define
 |
(3)
|
where the maximum is over integers 
relatively prime to 
,
and
 |
(4)
|
Thus 
is the largest absolute error for modulus 
, allowing both the endpoint 
and the reduced residue class 
to vary. The Bombieri-Vinogradov theorem then says that for
fixed 
,
 |
(5)
|
provided that 
.
In particular, for every 
, there is a 
such that
 |
(6)
|
Thus the theorem controls the average error for moduli up to essentially 
. It was proved independently by Bombieri (1965) and
A. I. Vinogradov (1965). The Elliott-Halberstam
conjecture proposes the corresponding estimate with 
replaced by 
for every 
.
-Series" [Russian]. Izv. Akad. Nauk SSSR Ser. Mat. 29,
903-934, 1965; corrigendum 30, 719-720, 1966.