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

Abstract

We establish a uniform approximation result for the Taylor polynomials of Riemann’s \(\xi \) 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 \(\xi \) 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 \(\xi \) function are also established. Finally, we explain how our approximation techniques can be extended to a collection of analytic L-functions.