A zero-free region for the fractional derivatives of the Riemann zeta function

  • Ricky E. Farr University of North Carolina
  • Sebastian Pauli University of North Carolina
  • Filip Saidak University of North Carolina
Keywords: Riemann zeta function, fractional derivatives, zero-free regions

Abstract

For any \alpha\in\R, we denote by D_s^{\alpha}[\zeta(s)] the α-th Grunwald-Letnikov fractional derivative of the Riemann zeta function ζ(s). For these derivatives we show:

D_s^\alpha[\zeta(s)]\ne 0

inside the region | s − 1 | < 1. This result, the first of its kind, is proved by a careful analysis of integrals involving Bernoulli polynomials and bounds for fractional Stieltjes constants.

 

Author Biographies

Ricky E. Farr, University of North Carolina

Department of Mathematics and Statistics

University of North Carolina

Greensboro, NC 27402, U.S.A.

Sebastian Pauli, University of North Carolina

Department of Mathematics and Statistics

University of North Carolina

Greensboro, NC 27402, U.S.A.

Filip Saidak, University of North Carolina

Department of Mathematics and Statistics

University of North Carolina

Greensboro, NC 27402, U.S.A.

Published
2020-09-04
Section
Articles