Difference between revisions of "Prime counting"

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The prime counting function $\pi \colon \mathbb{R} \rightarrow \mathbb{Z}^+$ is defined by the formula
 
The prime counting function $\pi \colon \mathbb{R} \rightarrow \mathbb{Z}^+$ is defined by the formula
 
$$\pi(x) = \{\mathrm{number \hspace{2pt} of \hspace{2pt} primes} \leq x \}.$$
 
$$\pi(x) = \{\mathrm{number \hspace{2pt} of \hspace{2pt} primes} \leq x \}.$$
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<div align="center">
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<gallery>
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File:Primecountingplot.png|Graph of $\pi(x)$.
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</gallery>
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</div>
  
 
=Properties=
 
=Properties=
<div class="toccolours mw-collapsible mw-collapsed" style="width:800px">
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[[Prime number theorem, pi and x/log(x)]]<br />
<strong>Theorem (Prime Number Theorem):</strong> The function $\pi(x)$ obeys the formula
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[[Prime number theorem, logarithmic integral]]<br />
$$\lim_{x \rightarrow \infty} \dfrac{\pi(x)}{\frac{x}{\log(x)}}=1.$$
 
<div class="mw-collapsible-content">
 
<strong>Proof:</strong>
 
</div>
 
</div>
 
  
 
=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