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

TL;DR

This paper studies the zeros of Taylor polynomials of the Riemann xi function, providing uniform approximation results, estimates of spurious zeros, and convergence behavior, with implications for understanding zeros of L-functions.

Contribution

It introduces a uniform approximation of the xi function's Taylor polynomials valid across the complex plane and analyzes the distribution and convergence of their zeros, extending to L-functions.

Findings

Taylor polynomials converge uniformly to the xi function in expanding domains

Number of spurious zeros outside the critical strip is estimated

Hurwitz zeros converge super-exponentially to xi function zeros

Abstract

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.