Riemann zeta function (idea)

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.)