In his famous paper of 1859, Riemann stated that the number of Riemann zeta function zeros with is asymptotically given by
(1)
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as (Edwards 2001, p. 19; Havil 2003, p. 203; Derbyshire 2004, p. 258). This can be written more compactly as
(2)
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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)
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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)
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Thus the mean spacing between and is
(5)
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which tends to zero as .