
Zeros and Poles of diagonal Pade Approximates to functions related to the Riemann Zeta functionGreg 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 (x1), 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. 