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Questions on various generalizations of the Riemann zeta function (e.g. Dedekind zeta, Hasse–Weil zeta, L-functions, multiple zeta). Consider using the tag (riemann-zeta) instead if your question is specifically about Riemann's function.

Riemann zeta function or, Euler–Riemann zeta function or, Zeta function in number theory are functions belonging to a class of analytic functions of a complex variable, comprising Riemann's zeta-function, its generalizations and analogues. Zeta-functions and their generalizations in the form of L-functions (cf. Dirichlet L-function) form the basis of modern analytic number theory. In addition to Riemann's zeta-function one also distinguishes the generalized zeta-function $~ζ(s,a)~$, the Dedekind zeta-function, the congruence zeta-function, etc.

Definition: The Riemann zeta function for $~s\in \mathbb{C}~$ with $~\operatorname{Re}(s)>1~$ is defined as $$\zeta(s)=\sum_{n=1}^{\infty}~\frac{1}{n^s}=1+\frac{1}{2^s}+\frac{1}{3^s}+\cdots$$ It is then defined by analytical continuation to a meromorphic function on the whole $\mathbb{C}$ by a functional equation.

Euler Product Representation: The Riemann zeta function for $~s\in \mathbb{C}~$ with $~\operatorname{Re}(s)>1~$ can be written as $$\zeta(s)=\prod_{p~\text{prime}}~(1-p^{-s})^{-1}$$

Applications: The Zeta function is an extremely important special function of mathematics and physics that arises in definite integration and is intimately related with very deep results surrounding the prime number theorem. While many of the properties of this function have been investigated, there remain important fundamental conjectures (most notably the Riemann hypothesis) that remain unproved to this day.

References:

https://en.wikipedia.org/wiki/Riemann_zeta_function

http://mathworld.wolfram.com/RiemannZetaFunction.html