Riemann Zeta Function

$$\zeta(s) = \sum_{k=1}^\infty {\frac{1}{k^s}}$$

Convergence of this function can be derived using integral test. The above-mentioned series is referred to as p-series.

Extension

The $\zeta$ function can be extended to the set $\mathbb{C}-\set{1}$. $$

Ramanujan Sum

$$\zeta(-1) = -\frac{1}{12}$$

Which is why, it’s used as below:

$$1+2+3+4+5+\dots = -\frac{1}{12}$$

This is known as the Ramanujan sum of the diverging series.

Riemann Hypothesis

The $\zeta$ function has its zeros only at negative even integers and complex numbers with real part $\frac{1}{2}$. $$

One of the most important unsolved problems in mathematics.

Euler Product formula

$$\zeta(z)=\prod_{n=1}^\infty \frac{1}{1-p_n^{-z}}$$

Where $p_n$ is the $n$-th prime number.