Prime counting
From specialfunctionswiki
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 $[0,1000000]$.
Contents
Properties
Theorem
The function $\pi(x)$ obeys the formula $$\lim_{x \rightarrow \infty} \dfrac{\pi(x)}{\frac{x}{\log(x)}}=1,$$ where $\pi$ denotes the prime counting function and $\log$ denotes the logarithm.
Proof
References
Theorem
The following formula holds: $$\lim_{x \rightarrow \infty} \dfrac{\pi(x)}{\mathrm{li}(x)}=1,$$ where $\pi$ denotes the prime counting function and $\mathrm{li}$ denotes the logarithmic integral.
