|
|
| Line 1: |
Line 1: |
| 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]].
| | Nice to meet you, my title is Ling and I totally dig that title. To perform badminton is some thing he truly enjoys performing. Arizona is her birth location and she will never transfer. Bookkeeping is how he supports his family and his wage has been really fulfilling.<br><br>Also visit my homepage: [http://Www.shownetbook.com/enone/xe/?document_srl=3029968 auto warranty] |
| | |
| 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 19:30, 5 February 2014
Nice to meet you, my title is Ling and I totally dig that title. To perform badminton is some thing he truly enjoys performing. Arizona is her birth location and she will never transfer. Bookkeeping is how he supports his family and his wage has been really fulfilling.
Also visit my homepage: auto warranty