Jensen polynomials for the Riemann zeta function and other sequences

• Type of Material: Article
• Authors: Michael Griffin, Brigham Young University; Ken Ono, Emory University; Larry Rolen, Vanderbilt University; Don Zagier, Max Planck Institute
• Language: English
• Date: 2019-06-04
• Publisher: National Academy of Sciences
• Publication Version: Final Published Version
• Copyright Statement: © 2019 the Author(s). Published by PNAS.
• License: Creative Commons Attribution-NonCommercial-NoDerivs 4.0 International
• Final Published Version (URL): http://dx.doi.org/10.1073/pnas.1902572116
• Title of Journal or Parent Work: PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
• Volume: 116
• Issue: 23
• Start Page: 11103
• End Page: 11110
• Grant/Funding Information: M.G. and K.O. were supported by NSF Grants DMS-1502390 and DMS-1601306. K.O. was supported by Asa Griggs Candler Fund.
• Keywords: hyperbolic polynomials; Jensen polynomials; Riemann hypothesis

Abstract

The Pólya–Jensen criterion for the Riemann hypothesis asserts that RH is equivalent to the hyperbolicity of certain Jensen polynomials for all degrees d≥1 and all shifts n. For each degree d≥1, we confirm this criterion for all sufficiently large shifts n. This represents a theoretical advance in the field. The method of proof is rooted in the newly discovered phenomenon that these polynomials are nicely approximated by Hermite polynomials. Furthermore, it is shown that this method applies to a large class of related problems.