## Dan Romik Studies the Riemann’s Zeta Function, and Other Zeta News.

Updates to previous posts: Karim Adiprasito expanded in a comment to his post on the g-conjecture on how to move from vertex-decomposable spheres to general spheres. Some photos were added to the post: Three pictures.

## Dan Romik on the Zeta function and its hypergeometric expansions

My friend Dan Romik wrote an impressive paper Orthogonal polynomial expansions for the Riemann xi function about expansion of the Riemann Zeta function. (I thank Dan for telling me about it.) Dan kindly agreed to write a blog post about his work in May 2019.

Dan Romik (click once to enlarge, twice to enlarge further)

## Michel Griffin, Ken Ono, Larry Rolen, and Don Zagier on Zeta and the hyperbolicity of Jensen polynomials in low degrees

Let me mention also another impressive paper by Michael Griffin, Ken Ono, Larry Rolen and Dan Zagier: Jensen polynomials for the Riemann zeta function and other sequences. (I learned about it from Ken Ono over Facebook. Ken writes: “After almost 2 years and 83 drafts, I am happy to share this paper on the Riemann Hypothesis. Amazingly, this paper has been recommended for publication 6 days after submission!” ). Here is a MO question regarding this development. (Update: There was a lot of media attention to the result. Here is a GLL post by R. J.  Lipton.)

## (Belated news) Polymath15 over Terry Tao’s blog on Noising and Denoising of the Riemann Zeta Function.

It is now the 10 year anniversary of polymath projects and let me belatedly mention a successful project polymath15 that took place over Terry Tao blog. The problem is related to the Hermite expansion of the Zeta function. Roughly speaking you can apply a noise operator on Zeta that causes higher degrees to decay exponentially. (The noise can be “negative” and then the higher degrees explode exponentially.) The higher the amount  t of noise the easier the assertion of the Riemann Hypothesis (RH) becomes. Let $\Lambda$, the de Bruijn-Newman constant, be the smallest amount of noise under which the assertion of RH is correct.  It was conjectured that $\Lambda \ge 0$ namely the assertion of the RH fails when you apply negative noise no matter how small. This follows from known conjectures on the spacing between zeroes of the Zeta function since when the assertion of RH is known for some level (positive or negative) of noise, then with a higher level of noise spacing between zeroes become more boring. The conjecture was settled by Brad Rodgers and Terry Tao. (See this blog post by Terry Tao.)

The task of polymath15 (proposed here, launched here, last (11th) post here) was to use current analytic number theory technology that already has yielded information on the zeroes of the zeta function (in the direction of RH and related conjectures) to deduce sharper upper bounds for $\Lambda$ than the previously known 0.5 value. The collaborative efforts led to

Theorem (Polymath 15): $\Lambda \le 0.22$

A remarkable success! (Of course, proving RH was not an objective of polymath15.) There were also  interesting comments of general nature regarding the RH and other big conjectures like this nice comment by anonymous.

It is even possible that $\Lambda =0$ (i.e. RH is true) and for each $t >0$ there is a simple Hermitian operator (possibly related to a probabilistic interpretation of $H_t$) having $H_t$ zeros as its spectrum (i.e. realizing Hilbert-Polya approach for $H_t$) but no such operator for $t=0$ (which may explain the failed attempts so far to find it). Since the zeros of $H_t$ converge to that of $H_0$, it is possible that there is a proof of RH via the functions $H_t$ (whose zeros are more regularly distributed) but not via direct attack on $H_0$ itself!
Such “homotopic approach” to study $H_0$ via $H_t$ reminds similar methods used in the past to solve big problems (e.g. de Branges solution – via Loewner’s PDE with flow parameter – of the Bieberbach conjecture, and Perelman’s solution – via Hamilton’s Ricci flow PDE – of the Poincare conjecture).

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### 4 Responses to Dan Romik Studies the Riemann’s Zeta Function, and Other Zeta News.

1. Jake says:

The link to the Dan Romik paper is broken ….

GK: corrected, thanks!

2. Gil Kalai says:

It is quite remarkable that wordpress’s (deep?) learning algorithm found a connection with the post “A Breakthrough by Maryna Viazovska Leading to the Long Awaited Solutions for the Densest Packing Problem in Dimensions 8 and 24”. Also Dan’s paper mentions works of Henry Cohn and of Maryna Viazovska.

3. Gil Kalai says:

I saw a very interesting paper by Keith Ball on rational approximations of the zeta function. https://warwick.ac.uk/fac/sci/maths/people/staff/keith_ball/rappcf.pdf