Table of Contents
1. Riemann Zeta Function
Analytic continuation of \[ \zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}. \]
2. Exact From of Prime Counting Function
Let us define \[ \pi_0(x) = \begin{cases} \pi(x) - \frac{1}{2},\quad & \text{if $x$ is prime}, \\ \pi(x),\quad & \text{otherwise}. \] Riemann proved that \[ \pi_0(x) = R(x) - \sum_{\rho \in \ker(\zeta)} R(x^{\rho}) \] where \[ R(x) := \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \mathrm{li} \left( x^{1/n} \right), \] \( \mu(n) \) is the Möbius function, \( \mathrm{li}(x) \) is the logarithmic integral function.
Other forms of \( R(x) \) include \[ R_{\rho = a+ bi}(x) = \mathrm{Re} \left( \sum_{n=1}^{\infty} \frac{\mu(n)}{n} \int_{-\infty + bi \log(x)/n}^{(a+bi) \log(x)/n} \frac{e^z}{z}\,\mathrm{d}z \right). \]
Assuming the Riemann hypothesis is true, the following can be deduced: \[ |\pi(x) - \mathrm{li}(x) | < \frac{\sqrt{x}\log(x)}{8\pi}.\]