View source for Riemann–von Mangoldt formula

{{no footnotes|date=October 2019}}

In [[mathematics]], the '''Riemann–von Mangoldt formula''', named for [[Bernhard Riemann]] and [[Hans Carl Friedrich von Mangoldt]], describes the distribution of the zeros of the [[Riemann zeta function]].

The formula states that the number ''N''(''T'') of zeros of the zeta function with imaginary part greater than 0 and less than or equal to ''T'' satisfies

:<math>N(T)=\frac{T}{2\pi}\log{\frac{T}{2\pi}}-\frac{T}{2\pi}+O(\log{T}).</math>

The formula was stated by Riemann in his notable paper "[[On the Number of Primes Less Than a Given Magnitude]]" (1859) and was finally proved by Mangoldt in 1905.

Backlund gives an explicit form of the error for all ''T'' greater than 2:
:<math>\left\vert{ N(T) - \left({\frac{T}{2\pi}\log{\frac{T}{2\pi}}-\frac{T}{2\pi} } - \frac{7}{8}\right)}\right\vert < 0.137 \log T + 0.443  \log\log T + 4.350 \ . </math>

==References==
* {{cite book | last=Edwards | first=H.M. | authorlink=Harold Edwards (mathematician) | title=Riemann's zeta function | series=Pure and Applied Mathematics | volume=58 | location=New York-London |publisher=Academic Press | year=1974 | isbn=0-12-232750-0 | zbl=0315.10035 }} 
* {{cite book | last=Ivić | first=Aleksandar | title=The theory of Hardy's ''Z''-function | series=Cambridge Tracts in Mathematics | volume=196 | location=Cambridge | publisher=[[Cambridge University Press]] | year=2013 | isbn=978-1-107-02883-8 | zbl=1269.11075}}
* {{cite book | last=Patterson | first=S.J. | title=An introduction to the theory of the Riemann zeta-function | series=Cambridge Studies in Advanced Mathematics | volume=14 | location=Cambridge | publisher=[[Cambridge University Press]] | year=1988 | isbn=0-521-33535-3 | zbl=0641.10029 }}

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[[Category:Theorems in analytic number theory]]
[[Category:Bernhard Riemann]]


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