Dynamic behavior of the roots of the Taylor polynomials of the Riemann xi function with growing degree

We establish a uniform approximation result for the Taylor polynomials of the xi function of Riemann which is valid in the entire complex plane as the degree grows. In particular, we identify a domain growing with the degree of the polynomials on which they converge to Riemann's xi function. Using this approximation we obtain an estimate of the number of "spurious zeros" of the Taylor polynomial which are outside of the critical strip, which leads to a Riemann - von Mangoldt type of formula for the number of zeros of the Taylor polynomials within the critical strip. Super-exponential convergence of Hurwitz zeros of the Taylor polynomials to bounded zeros of the xi function are established along the way, and finally we explain how our approximation techniques can be extended to a collection of analytic L-functions.