Prime counting function

The prime counting function, denoted $\pi$, is a function defined on real numbers. The quantity $\pi(x)$ is defined as the number of positive prime numbers less than or equal to $x$. Gauss first conjectured that the prime number theorem $\sim$ $\frac{x}{log x}$, or equivalently, $\lim_{x\to\infty}\frac{\pi(x)}{\frac{x}{log x}} = 1$.

The function $\pi(x)$ is asymptotically equivalent to $\frac{x}{log x}$. This is the prime number theorem. It is also asymptotically equivalent to Chebyshev's theta function. It was first proved in 1896 by Jacques Hadamard and by Charles de la Vallée Poussin, working independently.

The prime counting function has many ties to the Riemann zeta function and other branches of analytic number theory. For example, showing that $\pi(x) -$ $\int^n_2 \frac{1}{log x} =\mathcal{O}(\sqrt x log x)$, where $\mathcal{O}$ is Big O Notation, would imply the Riemann Hypothesis. However, this is a result stronger than the hypothesis, so to prove the Riemann Hypothesis, it would suffice to show that $\pi(x) -$ $\int^n_2 \frac{1}{log x} =\mathcal{O}(x^{\frac{1}{2}+\epsilon})$. This second statement is equivalent to the hypothesis.