Behavior of the Roots of the Taylor Polynomials of Riemann’s ξ Function with Growing Degree

We establish a uniform approximation result for the Taylor polynomials of Riemann’s ξ function 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 ξ function. Using this approximation, we obtain an estimate of the number of “spurious zeros” of the Taylor polynomial that lie outside of the critical strip, which leads to a Riemann–von Mangoldt type 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 ξ function are also established. Finally, we explain how our approximation techniques can be extended to a collection of analytic L -functions.