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

Robert Jenkins, Ken D.T.R. McLaughlin

Research output: Contribution to journalArticlepeer-review

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 languageEnglish (US)
Pages (from-to)265-293
Number of pages29
JournalConstructive Approximation
Volume49
Issue number2
DOIs
StatePublished - Apr 15 2019
Externally publishedYes

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Mathematics(all)
  • Computational Mathematics

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