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.

Interestingly, one of the applications of the zeta function is in superstring theory. If I remember correctly, the value of zeta(n) at n=-1 had something to do with the spacetime having dimension 26 for bosonic string theory.

If you look at the first definition for the Zeta function given above, you will notice that given an argument of -1, the function produces a sum of the positive integers from 1 to infinity (1 + 2 + 3 + 4 + ... to infinity!)

Zeta(-1) = -1/12

I am not making this up.

Euler was able to prove that

             ∞
           ____
           \     1
Zeta(z):=   \  ----
            /    z
           /___ m
            m=0

is equal to

  _____    1
   | |  -------
   | |      -z
   | |   1-p
   p∈P

for all p in P, the prime numbers. Thus the equivalent expression is

    1        1        1
--------*--------*-------- . . .
1-2^(-z) 1-3^(-z) 1-5^(-z)

where z can be any complex number. This function reputedly evaluates to zero for z=-2, -4, -6, . . ., as a quick check will confirm. Riemann suspected that this function would evaluate to zero for complex numbers z=a+bi only when a=-2n and b=0 (trivial cases) or when a is near (equal to?) 1/2 and b is some nonzero value. This is called the Riemann Hypothesis and is as yet an open question in mathematics.

It is a matter of complex algebra to show that the above multiplication can be rearranged to the following form (where z=a+bi):

             _____                1-p^(-a+bi)
              | |   ------------------------------------
Zeta(a+bi) =  | |   sqrt(1-2*p^(-a)*cos(bln(p))+p^(-2a))
              | | ----------------------------------------
              p∈P   sqrt(1-2*p^(-a)*cos(bln(p))+p^(-2a))

In this form, the complex numerator (include the very top and middle terms, i.e., the 'fraction on top') is always a unit in "vector length" (and thus causes no "vector scaling"), and the "real" denominator (the very bottom sqrt term) is the only thing that affects the possibility of convergence. (Yes, I agree that it's relatively ugly. The point is to separate real and complex, and only give the real the chance at converging.)

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