# Riemann Zeta function

The Riemann Zeta function, commonly denoted ζ(s) is a very important function in number theory. The function can be defined as $\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}$ when s has a real part greater than 1. The function can then be analytically continued for everywhere where s is non-one (the function has a simple pole at s = 1). The function was investigated by a number of mathematicians including Euler who showed that $\zeta(2) = \frac{\pi^2}{6}$ but was first extensively investigated by Bernhard Riemann who showed that the behavior of the function was intimately connected to the distribution of prime numbers. In particular, where the function is zero is related very closely to the error term of the Prime Number Theorem and the prime counting function. The Riemann hypothesis is an unsolved conjecture that describes strong restrictions on the locations of the zeros (specifically, that all non-trivial zeroes are complex numbers of the form z = 1/2 + i y).

The convergence of the Riemann zeta function can be extended over all complex numbers first by stretching to converge over s with real part greater than zero,

$\zeta(s) = \left(1-\frac{1}{2^{s-1}}\right)\sum_{n=1}^\infty (-1)^{n+1}\frac{1}{n^s}$

and then to all complex numbers via,

$\zeta(1-s)=2^{1-s}\pi^{-s}\sin\left(\frac{1-s}{2}\pi\right)\Gamma(s-1)\zeta(s)$.

Two thing are noticeable from the last equation, the function is zero at negative even integers (the trivial zeros) and that the function has dependence on the values on either side of the real line 1/2.

## Relation to the Prime Numbers

The zeta function is frequently said to "encode" the prime numbers in its structure. This was first noticed in conjunction with the Prime Number Theorem, with the realization that the derivative of the zeta function, divided by the zeta function itself, could be expressed in a power series

$\frac{\zeta'(z)}{\zeta(z)} = \sum_{n=1}^\infty \frac{\Lambda(n)}{n^z} \$

where $\Lambda(n) = \begin{cases} \log p, & \mbox{if } n = p^m, m \in \mathbb{N} \\ 0, & \mbox{otherwise } \end{cases}$ and $p \$ is a prime number.

A more specific relation was later proved, that the number of prime numbers less than some value $x \$, $\pi(x) \$, is precisely related to a sum of sums of functions involving the zeroes of the zeta function. [1]