This post, about the Riemann zeta function, which is among the most important and mysterious mathematical objects was kindly written by Dan Romik. It is related to his paper Orthogonal polynomial expansions for the Riemann xi function, that we mentioned in this post.
Dan Romik on the Riemann zeta function
Recently when I was thinking about the Riemann zeta function, I had the double thrill of discovering some new results about it, and then later finding out that my new ideas were closely related to some very classical ideas due to two icons of twentieth-century mathematics, George Pólya and Pál Turán. When you are trying to stand on the shoulders of giants, it’s nice to see other giants right there beside you trying to do the same!
It all goes back to one of the most famous problems in mathematics, the Riemann Hypothesis (RH). Both Pólya and Turán were rather enamored with this problem and published about it extensively; Pólya was said to have been preoccupied with the problem to the very end of his life.(1) And they both recognized that an important first step in trying to prove something about the zeros of the zeta function is having a good representation
for the Riemann zeta function. After all, there are many different formulas that can be used to define or compute the zeta function. If you don’t choose the right one, you probably won’t get very far with your analysis.
Pólya in one of his famous attacks on the problem considered the representation of the zeta function (or more precisely of the Riemann xi function, which is a symmetrized and better-behaved version of the zeta function; see below) as a Fourier transform—a standard representation due (essentially) to Riemann. I’ll have more to say about that later.
Turán also looked at the Riemann xi function, and instead of working with one of the standard “named” representations such as the Fourier transform or Taylor series, looked around a bit more intentionally for a representation of the function that seemed particularly suited to answering the specific question of whether the zeros all lie on a line. In a 1950 address to the Hungarian Academy of Sciences, he put forward his ideas about what he thought was the correct representation to look at: the infinite series expansion of the xi function in the Hermite polynomials. About eighty years after Turán’s discovery, my own investigations led me to discover  that the Hermite polynomials are not the only polynomials in which it’s interesting to expand the Riemann xi function. It turns out that there are at least two other families of polynomials for which the respective expansions are no less (and, in some ways, more) well-behaved. My motto for these polynomial families, which are known to experts in special functions but have until now been somewhat esoteric (though I hope that is about to change), is that they are “the coolest polynomials that you never heard of.”
Let’s look at some of the technical details so that I can explain why these new expansions are interesting, and how they relate to Turán’s work and ultimately back to Pólya’s ideas and one of the particular threads that grew out of them. First, define the Riemann xi function as
where is the Euler gamma function and is the Riemann zeta function. It’s also common to denote
This is Riemann’s “capital xi” function, which is still usually referred to as Riemann’s xi function. (This seems reasonable: the two functions are the same up to a trivial linear change of variables.) The main point of these definitions is that is an entire function of the complex variable , and that RH can now be reformulated as the statement that the zeros of all lie on the real line. Moreover, the famous functional equation satisfied by the Riemann zeta function maps to the statement that is an even function.
Now consider the following four ways of representing the xi function:
Here, the first representation (1) is simply the Taylor expansion of , which contains only even terms since is an even function. The numbers are (up to the sign factor) the Taylor coefficients. Some attempts have been made to understand them, and one interesting and fairly trivial observation (again going back to facts already known to Riemann) is that they are all positive. Some additional and less trivial things can be said—see for example Section 6.1 of my paper , and the recent paper by Griffin, Ono, Rolen and Zagier . But at the end of the day, no one has yet succeeded in using the Taylor expansion to prove anything new about the location of the zeros.
The second representation (2) is the infinite series expansion of in the classical sequence of Hermite polynomials, defined by the well-known formula
This is the representation whose use was advocated by Turán. His reasoning was that expanding a function of a complex variable (for example, in the simplest case, a polynomial) in monomials doesn’t provide useful information to easily decide if the function has only real zeros, because the monomials have, roughly speaking, radial symmetry: their level curves are concentric circles. The Hermite polynomials on the other hand, at least heuristically, have level curves that are closer to being straight lines parallel to the real axis, Turán argued; thus, they are more suited to the geometry of the problem we are trying to solve.
Turán’s case for supporting the Hermite polynomials as the right basis to use is quite detailed—you can read about it in his papers [6,7,8] (and no, he was not able to actually prove anything about the zeros of ; this is a common theme in most of the attacks on RH to date…). I’ll simply mention that again one interesting and fairly easy observation is that the coefficients in the expansion (2)—adjusted through the introduction of the sign factor —end up being positive numbers. Their asymptotic behavior can also be analyzed: I prove a result about this in my paper (though it’s not particularly pretty).
Now comes the part that to me seems the most exciting, involving the expansions (3) and (4). These are the expansions in the more exotic families of polynomials
(where is a Pochhammer symbol), mildly rescaled by replacing with . In the terminology of the theory of orthogonal polynomials, the family is a special case of a two-parameter family known as the Meixner-Pollaczek polynomials, with the parameters taking the particular values . Similarly, the family is a special case of the four-parameter family known as the continuous Hahn polynomials, with the parameters taking the particular values . Their main characterizing property is that they are orthogonal sequences of polynomials for two specific weight functions on : the are orthogonal with respect to the weight function , and the are orthogonal with respect to . Again, fairly esoteric. But interesting!
There are several things that make the expansions (3)–(4) well-behaved. First, the coefficients , are again positive. This actually seems pretty relevant for questions like RH: for example, if we consider “toy” versions of (1)–(3) in which the coefficient sequences , and are replaced by the sequence for fixed , all three expansions sum up to rescaled cosines, which are entire functions that of course have only real zeros. (Without the factor, we would get a hyperbolic cosine, which has imaginary zeros.)
Second, one can derive asymptotics for and , and they are quite a bit nicer than the asymptotic formulas for the Taylor and Hermite expansion coefficients. In my paper, I proved that where are the constants
Third, the expansions have some conceptual meaning: (3) turns out to be equivalent to the expansion of the elementary function , , in an orthogonal basis of functions related to the Laguerre polynomials . And analogously, (4) arises out of the expansion of a certain auxiliary function (I won’t define it here) in yet another classical family of orthogonal polynomials, the Chebyshev polynomials of the second kind.
Fourth (and fifth, sixth, …): the expansions are just… nice, in the sense that they arise in a way that seems natural when one asks certain questions, that they have excellent convergence properties, and that the coefficients and have several elegant formulas, each revealing something interesting about them. Read the paper to understand more.
I said I will get back to Pólya’s work on RH. This post is already quite long so I will say only a little bit about this. One of Pólya’s major discoveries was that there are operations on entire functions that (under certain mild assumptions) preserve the property of the function having only real zeros. Specifically this is the case for the operation of multiplying the Fourier transform of the function by the factor for (where is the frequency variable). This opens the way to defining a family of deformations of the Riemann xi function arising out of this operation, and trying to generalize RH by asking for which values of it is the case that has only real zeros. Since Pólya’s work, and important later extensions of it by De Bruijn and Newman, this has become a very active topic of research, nowadays referred to under the name of the De Bruijn-Newman constant.
See the recent survey of Newman and Wu , a 2018 paper by Rodgers and Tao  proving a major conjecture of Newman, and the recent paper  by the Polymath15 project (mentioned by Gil in his earlier post), for the latest on this subject.
The connection I found between this topic and the idea of expanding the Riemann xi function in families of orthogonal polynomials is the following: expansions such as (2)–(4) suggest yet another natural way of “deforming” the Riemann xi function by adding a parameter : simply multiply the th term in the expansion by (the linear operator that does this is called the Poisson kernel, and generalizes the standard Poisson kernel from complex analysis and the theory of harmonic functions). It turns out—and is actually easy to prove, and really isn’t terribly surprising in the grand scheme of things—that in the case of the Hermite expansion (2), this family of deformations is the same, up to some trivial reparametrization, as the family of deformations that was studied in connection with the work of Pólya, De Bruijn, Newman and their successors. A nice connection between two threads of research that were not previously recognized as being related to each other, I think. Furthermore, this suggests that the Poisson kernel and associated deformations may yet have an important role to play in the context of the new expansions in the orthogonal polynomial families and , where we get genuinely new families of deformations of the Riemann xi function. I explore this idea in my paper and it leads to some interesting things.
So let’s summarize. The key questions you are no doubt wondering about are: where does any of this lead? And do these new ideas say anything really useful or especially relevant for the Riemann hypothesis? The answer is that I don’t know (and I’m wondering about the same things). That being said, these orthogonal polynomial expansions seem quite interesting in their own right. The Riemann zeta function is a mysterious object, and there are other things we wish to understand about it beside where its zeros are, so it’s always good to have additional points of view from which to approach it. Moreover, even on the question of the zeros there are reasons to be cautiously optimistic that this approach may have something useful to offer; see Chapter 7 of my paper for a brief discussion of why that is the case.
 D. Albers and G. L. Alexanderson, editors. Mathematical People: Profiles and Interviews. A K Peters, 2008.
 M. Griffin, K. Ono, L. Rolen and D. Zagier. Jensen polynomials for the Riemann zeta function and other sequences. Preprint (2019), arXiv:1902.07321.
 C. M. Newman. Constants of de Bruijn-Newman type in analytic number theory and statistical physics. To appear in Bull. Amer. Math. Soc.
 B. Rodgers and T. Tao. The De Bruijn-Newman constant is nonnegative. Preprint (2018), arXiv:1801.05914.
 D. Romik. Orthogonal polynomial expansions for the Riemann xi function. Preprint (2019), arXiv:1902.06330.
 P. Turán. Sur l’algèbre fonctionelle. Pages 279–290 in: Comptes Rendus du Premier Congrès des Mathématiciens Hongrois, 27 Août–2 Septembre 1950. Akadémiai Kiadó, 1952. Reprinted in Collected Papers of Paul Turán, Ed. P. Erdos, Vol. 1, pp. 677–688. Akadémiai Kiadó, 1990. An English translation of the paper by Dan Romik On functional algebra.
 P. Turán. Hermite-expansion and strips for zeros of polynomials. Arch. Math. 5 (1954), 148–152. Reprinted in Collected Papers of Paul Turán, Ed. P. Erdos, Vol. 1, pp. 738–742. Akadémiai Kiadó, 1990.
 P. Turán. To the analytical theory of algebraic equations. Bulgar. Akad. Nauk. Otd. Mat. Fiz. Nauk. Izv. Mat. Inst. 3 (1959), 123–137. Reprinted in Collected Papers of Paul Turán, Ed. P. Erdos, Vol. 2, pp. 1080–1090. Akadémiai Kiadó, 1990.
 D.H.J. Polymath. Effective approximation of heat flow evolution of the Riemann ξ function, and a new upper bound for the de Bruijn-Newman constant. Preprint (2019), arXiv:1904.12438.
(1) Alexanderson writes in [1, p. 259]: “A week or so before he died, Pólya asked me to look on his desk at home for some papers on which he said he had written down some interesting ideas he had for proving RH. Of course I could find no such notes, but until the day he died he was thinking about that famous problem.”
(2) Turán’s Hungarian Academy of Sciences talk was published in a rather obscure French-language paper  that seems to have been largely forgotten. It’s an interesting read nonetheless, and to make it more accessible to anyone who may be interested, I recently translated it to English.
(3) Turán mentions in  that he discovered the results on the Hermite expansion in 1938–39, but they were not published until much later. Clearly this was not a convenient time in history for publishing such discoveries; Turán, a Hungarian Jew, spent much of World War II interned in labor camps in Hungary.