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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 famous paper ''[[On the Number of Primes Less Than a Given Magnitude]]'' (1859) and proved by [[von 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=pre06093527 }} | |||
* {{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 }} | |||
{{DEFAULTSORT:Riemann-von Mangoldt formula}} | |||
[[Category:Analytic number theory]] | |||
[[Category:Theorems in number theory]] | |||
{{numtheory-stub}} | |||
Revision as of 17:12, 30 October 2013
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
The formula was stated by Riemann in his famous paper On the Number of Primes Less Than a Given Magnitude (1859) and proved by von Mangoldt in 1905.
Backlund gives an explicit form of the error for all T greater than 2:
References
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My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
My blog: http://www.primaboinca.com/view_profile.php?userid=5889534