Proceedings of the National Academy of Sciences· 2019Q1
Jensen polynomials for the Riemann zeta function and other sequences
- 115citations
- Q1SCImago
- 2019year
Short summary
Jensen polynomials for the Riemann zeta function are hyperbolic for all but finitely many degrees, proving a key conjecture and a prediction from random matrix theory.
AI-generated from the title and abstract; the full text is not read.
Key points
- Proves hyperbolicity for all but finitely many Jensen polynomials of each degree for the Riemann zeta function.
- Establishes hyperbolicity for all Jensen polynomials up to degree 10.
- Develops a general theorem modeling Jensen polynomials with Hermite polynomials.
- Confirms a prediction from the Gaussian unitary ensemble random matrix model.
- Proves a conjecture by Chen, Jia, and Wang on the partition function.
AI-generated from the title and abstract; the full text is not read.
Abstract
In 1927, Pólya proved that the Riemann hypothesis is equivalent to the hyperbolicity of Jensen polynomials for the Riemann zeta function [Formula: see text] at its point of symmetry. This hyperbolicity has been proved for degrees [Formula: see text] We obtain an asymptotic formula for the central derivatives [Formula: see text] that is accurate to all orders, which allows us to prove the hyperbolicity of all but finitely many of the Jensen polynomials of each degree. Moreover, we establish hyperbolicity for all [Formula: see text] These results follow from a general theorem which models such polynomials by Hermite polynomials. In the case of the Riemann zeta function, this proves the Gaussian unitary ensemble random matrix model prediction in derivative aspect. The general theorem also allows us to prove a conjecture of Chen, Jia, and Wang on the partition function.
The authors' abstract, as published at the source. Proceedings of the National Academy of Sciences, 2019 · DOI ↗
Continue with a free account
Ask the paper: 3 free questions a day about this paper; save it, get its citation, new summaries every day for your field. Takeaways are Premium.
Continue free on the webSign in with Google or Apple; no card needed. You come back to this paper.
On your phone:
Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics