Difference between revisions of "Prime counting"
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<strong>Theorem (Prime Number Theorem):</strong> The function $\pi(x)$ obeys the formula | <strong>Theorem (Prime Number Theorem):</strong> The function $\pi(x)$ obeys the formula | ||
| − | $$\lim_{x \rightarrow \infty} \dfrac{\pi(x)}{\frac{x}{\log(x)}}.$$ | + | $$\lim_{x \rightarrow \infty} \dfrac{\pi(x)}{\frac{x}{\log(x)}}=1.$$ |
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<strong>Proof:</strong> | <strong>Proof:</strong> | ||
Revision as of 19:03, 19 October 2014
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
Theorem (Prime Number Theorem): The function $\pi(x)$ obeys the formula $$\lim_{x \rightarrow \infty} \dfrac{\pi(x)}{\frac{x}{\log(x)}}=1.$$
Proof:
