The Reimann Zeta function is the most well known of many zeta functions, being originally defined by the infinite sum:
Zeta(n):=   \    1
            /   ---
           /__  m^n

However, it was analytically extended into the argand plane to be defined as:
          /\ oo
          |      u^(z-1)
           \    --------- du
           |     e^u - 1
          \/ 0
Zeta(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.