### Abstract

We establish a uniform approximation result for the Taylor polynomials of Riemann’s (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) 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) | 1-29 |

Number of pages | 29 |

Journal | Constructive Approximation |

DOIs | |

State | Accepted/In press - Feb 9 2018 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Mathematics(all)
- Computational Mathematics

### Cite this

**Behavior of the Roots of the Taylor Polynomials of Riemann’s ξ Function with Growing Degree.** / Jenkins, Robert; Mclaughlin, Kenneth D T.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Behavior of the Roots of the Taylor Polynomials of Riemann’s ξ Function with Growing Degree

AU - Jenkins, Robert

AU - Mclaughlin, Kenneth D T

PY - 2018/2/9

Y1 - 2018/2/9

N2 - We establish a uniform approximation result for the Taylor polynomials of Riemann’s (Formula presented.) 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 (Formula presented.) 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 (Formula presented.) function are also established. Finally, we explain how our approximation techniques can be extended to a collection of analytic L-functions.

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

KW - Hurwitz zeros

KW - Riemann zeta

KW - Szegő curves

KW - Taylor polynomials

UR - http://www.scopus.com/inward/record.url?scp=85041828618&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85041828618&partnerID=8YFLogxK

U2 - 10.1007/s00365-018-9417-7

DO - 10.1007/s00365-018-9417-7

M3 - Article

AN - SCOPUS:85041828618

SP - 1

EP - 29

JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

ER -