Zeros and Poles of diagonal Pade Approximates to functions related to the Riemann Zeta function
Greg Fee, CECM
Abstract: The Riemann Zeta function is defined by: Zeta(x) = sum(k^(-x),k=1..infinity). By using analytic continuation we can extend the definition of this function from real x>1 to the entire complex plane, except for a simple pole at x=1. We may remove the pole by subtracting it from the function or by multiplying the Zeta function by (x-1), or we can use Riemann's symmetric Zeta function. First we compute approximate truncated Taylor series of the above functions expanded about the points 0 or 1/2 or 1. Next we compute diagonal Pade approximates to each of the Taylor series. Then we find the zeros and poles of these Pade approximates.