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"> | ||
| + | <gallery> | ||
| + | File:Primecountingfunction.png|Plot of $\pi(x)$ over $[0,50]$. | ||
| + | File:Primecountingfunctiondividedbyxoverlogx.png|Plot of $\frac{\pi(x)}{x/\log(x)}$ on $[0,1000000]$. | ||
| + | </gallery> | ||
| + | </div> | ||
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=Properties= | =Properties= | ||
Revision as of 06:31, 5 April 2015
The prime counting function $\pi \colon \mathbb{R} \rightarrow \mathbb{Z}^+$ is defined by the formula $$\pi(x) = \{\mathrm{number \hspace{2pt} of \hspace{2pt} primes} \leq x \}.$$
- Primecountingfunction.png
Plot of $\pi(x)$ over $[0,50]$.
- Primecountingfunctiondividedbyxoverlogx.png
Plot of $\frac{\pi(x)}{x/\log(x)}$ on $[0,1000000]$.
Properties
Theorem (Prime Number Theorem): The function $\pi(x)$ obeys the formula $$\lim_{x \rightarrow \infty} \dfrac{\pi(x)}{\frac{x}{\log(x)}}=1.$$
Proof:
