The Reimann Zeta function is the most well known of many zeta functions, being originally defined by the infinite sum:

oo
__
\
Zeta(n):= \ 1
/ ---
/__ m^n
m=0

However, it was analytically extended into the argand plane to be defined as:

/\ oo
| u^(z-1)
\ --------- du
| e^u - 1
\/ 0
Zeta(z):= -------------------
Gamma(z)

This relationship is responsible for the

trivial roots at -2, -4, -6, ... however there are infinitely many more

non-trivial roots in the argand plane. The

Riemann Hypothesis deals with their location.
The Riemann Zeta function has special names and values for certain values of n or z: at 1 the series formed is called the harmonic series; at 2 this equals

pi^2/6; at 3 is an

irrational, possibly

transcendental number called

Apery's Constant; at positive, even values it is a constant times a power of pi, and at odd the values are so far unknown. The harmonic series diverges very slowly and so is a

simple pole with a

residue of 1.