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# New progress on the zeta function: From old conjectures to a major breakthrough

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## Early History

The zeta function

The paper by Griffin et al. (1) makes fundamental progress in the study of the Riemann zeta function by introducing a method to study certain classical polynomials (the so-called Jensen polynomials) that were known to play a role for understanding the finer properties of the zeta function but had proved to be quite intractable to study by means of standard methods. What was known before this work was a plausible but inaccessible conjecture, called hyperbolicity, for all of them. In this paper the authors introduce a method to study these polynomials which allows the authors to establish hyperbolicity for a very large subset of them.

As an example of the progress made, the random matrix model for the zeta and allied functions was proposed in a fundamental paper by Keating and Snaith (2), putting aside many previous naive attempts to use a Gaussian law to explain the randomness of arithmetical functions. Consider for example tossing coins with the function

This paper rigorously shows that random matrix theory is needed to explain the oscillations of the classic zeta function. In the geometric setting of arithmetic over fields of characteristic p, this was proved to be the case in a famous work by Katz and Sarnak (3). At last, we have now another big step in the right direction and a further vindication of the insight of mathematical physicists. Undoubtedly, this paper will be of interest to a public going beyond number theorists.

The behavior of

The conjecture of Jensen is that the roots of the polynomials associated to the Taylor expansion of

Generalized Jensen polynomials show up in various other contexts, particularly in convex optimization, but only relatively recently have they, and their extension to a theory in several variables, attracted real attention [see, e.g., Bauschke et al. (6)].

## The Main Result

In this relatively short paper, the authors are able to prove the desired hyperbolicity for a big chunk of the original Jensen polynomials, namely for every fixed degree d and all

There is no doubt that this paper will inspire further fundamental work in other areas of number theory as well as in mathematical physics.

## Footnotes

- ↵
^{1}Email: eb{at}math.ias.edu.

The author declares no conflict of interest.

Author contributions: E.B. wrote the paper.

See companion article on page 11103.

Published under the PNAS license.

## References

- ↵
- M. Griffin,
- K. Ono,
- L. Rolen,
- D. Zagier

- ↵
- ↵
- N. M. Katz,
- P. Sarnak

- ↵
- G. Pólya

- ↵
- M. Chasse

- ↵
- H. H. Bauschke,
- O. Güler,
- A. S. Lewis,
- H. S. Sendov

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