In his famous paper of 1859, Riemann stated that the number 
of Riemann zeta
function zeros 
with 
is asymptotically given by
 |
(1)
|
as 
(Edwards 2001, p. 19; Havil 2003, p. 203; Derbyshire 2004, p. 258).
This can be written more compactly as
 |
(2)
|
This result was proved by von Mangoldt in 1905 and is hence known as the Riemann-von Mangoldt formula.
It follows that the density 
of zeros at height 
is
 |
(3)
|
where, as usual, the asymptotic notation 
means that the ratio 
tends to 1 as 
.
Another consequence of this result is that the imaginary parts of consecutive zeta zeros in the upper half-plane 
satisfy
 |
(4)
|
Thus the mean spacing 
between 
and 
is
 |
(5)
|
which tends to zero as 
.
