For \(x \in \mathbb N\), \begin{equation} \pi(n) = \sharp\, \{p \leq n \mid p \text{ is prime}\}. \end{equation} Then \begin{equation} \pi(x) \sim \int_2^{x} \frac{1}{\ln t}dt \equiv \text{Li}(x), \end{equation} which is the prime number theory. The concept map and proof process can be seen from the graph: Concept Map. A little bit work shows then \begin{equation} \pi(x) \sim \frac{x}{\ln x}. \end{equation} But \(\text{Li}(x)\) is a better approximation to \(\pi(x)\) than \(\frac{x}{\ln x}\) is. RH \(\equiv\) Riemann Hypothesis (approximately): For any \(\epsilon > 0\), \begin{equation} |\pi(x) - \text{Li}(x)| = O\bigl(x^{\frac{1}{2} + \epsilon}\bigr). \end{equation} More careful analysis shows that \(\text{Li}(x) = \frac{x}{\ln x} + \frac{x}{(\ln x)^{2}} + O\bigl(\frac{x}{(\ln x)^{3}}\bigr)\). So if RH is true, \begin{equation} |\pi(x)-\frac{x}{\ln x}| \sim \frac{x}{(\ln x)^{2}}. \end{equation}