## Abstract

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.

Original language | English (US) |
---|---|

Pages (from-to) | 265-293 |

Number of pages | 29 |

Journal | Constructive Approximation |

Volume | 49 |

Issue number | 2 |

DOIs | |

State | Published - Apr 15 2019 |

Externally published | Yes |

## Keywords

- Hurwitz zeros
- Riemann zeta
- Szegő curves
- Taylor polynomials

## ASJC Scopus subject areas

- Analysis
- Mathematics(all)
- Computational Mathematics