Behavior of the Roots of the Taylor Polynomials of Riemann’s \(\xi \) Function with Growing Degree

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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.

Keywords

Riemann zeta Taylor polynomials Szegő curves Hurwitz zeros 

Mathematics Subject Classification

30E10 11M26 

Notes

Acknowledgements

The authors thank Nigel Pitt for useful conversations. R. Jenkins was supported by the National Science Foundation under grant DMS-1418772. K. D. T-R McLaughlin was supported by the National Science Foundation under grant DMS-1401268.

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ArizonaTucsonUSA
  2. 2.Department of MathematicsColorado State UniversityFort CollinsUSA

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