Prime counting

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:

References

Newman's short proof of the prime number theorem