Difference between revisions of "Prime counting"

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(Properties)
 
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File:Primecountingfunction.png|Plot of $\pi(x)$ over $[0,50]$.
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File:Primecountingplot.png|Graph of $\pi(x)$.
File:Primecountingfunctiondividedbyxoverlogx.png|Plot of $\frac{\pi(x)}{x/\log(x)}$ on $[0,1000000]$.
 
 
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=Properties=
 
=Properties=
{{:Prime number theorem, pi and x/log(x)}}
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[[Prime number theorem, pi and x/log(x)]]<br />
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[[Prime number theorem, logarithmic integral]]<br />
  
 
=References=
 
=References=
 
[http://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2975232/fulltext.pdf Newman's short proof of the prime number theorem]
 
[http://people.mpim-bonn.mpg.de/zagier/files/doi/10.2307/2975232/fulltext.pdf Newman's short proof of the prime number theorem]
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[[Category:SpecialFunction]]
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{{:Number theory functions footer}}

Latest revision as of 06:35, 22 June 2016

The prime counting function $\pi \colon \mathbb{R} \rightarrow \mathbb{Z}^+$ is defined by the formula $$\pi(x) = \{\mathrm{number \hspace{2pt} of \hspace{2pt} primes} \leq x \}.$$

Properties

Prime number theorem, pi and x/log(x)
Prime number theorem, logarithmic integral

References

Newman's short proof of the prime number theorem

Number theory functions