Positivity of the coefficients of Taylor series associated to the Riemann hypothesis

The question below relates to the paper "Jensen Polynomials for the Riemann Zeta Function and Other Sequences" of Griffin, Ono, Rolen and Zagier. I'm asking it here because I am sure the answer is well-known to a specialist, and was unsuccessful getting an answer at MSE. All definitions included below can be found in the above paper.

Recall that George Pólya proved that the Riemann Hypothesis (RH) is equivalent to the hyperbolicity of the Jensen polynomials associated to the sequence of Taylor coefficients $$\{\gamma(n)\}$$ defined by \begin{align} (-1+4z^{2})\Lambda(\frac{1}{2}+z)=\sum_{n=0}^{\infty}\frac{\gamma(n)}{n!}z^{2n}, \nonumber \end{align} where $$\Lambda(s):=\pi^{-s/2}\Gamma(\frac{s}{2})\zeta(s)$$.

There is a table in the above paper containing some values of $$\gamma(n)$$ for large $$n$$, and all of these values are positive (albeit only slightly so).

Question: Is it known whether $$\gamma(n)>0$$ for all $$n\in\mathbb{N}$$?

In case it is helpful, there are formulas for these beasts, when $$n$$ is positive:

$$\displaystyle\gamma(n)=\frac{n!}{(2n)!2^{2n-1}}\left(32{2n \choose 2}F(2n-2)-F(2n)\right)$$

where $$\displaystyle F(n)=\int_{1}^{\infty}\ln(t)^{n}t^{-3/4}\theta_{0}(t)\mathrm{d}t$$

and the theta function $$\displaystyle \theta_{0}(t)=\sum_{k=1}^{\infty}e^{-\pi k^{2}t}$$.

(I expect the answer is something like "Of course they aren't all positive, by the result of the paper you link to...it's because they are modeled eventually by Hermite Polynomials which themselves don't have this property", but I can't see this clearly, so am hoping for expert clarification. Thanks, in advance!)

• Just curious - there seems to be some discrepancy with the constant term. It should be $-\pi^{-1/4}\Gamma(1/4)\zeta(1/2)\approx3.97697$, while your formula gives, if one assumes $0\times F(-2)=0$, something like $-0.023$. Presumably analytic continuation of $F$ has a pole at $-2$ and one needs to compute certain limit to obtain correct value for $\gamma(0)$? Not that it is terribly important, but... Apr 24 at 19:06
• @მამუკაჯიბლაძე: I found this formula for $\gamma(n)$ in the same paper. It's formula [13], in case this helps, and you are right...these formulas require $n$ to be positive. Apr 24 at 19:08
• "I'm asking it here because I am sure the answer is well-known to a specialist, and was unsuccessful getting an answer at MSE." I can't find any question of yours on this topic at math.stack. Has it been deleted? Jun 3 at 23:52
• @GerryMyerson: yes, I've deleted it upon request of users over at MSE. Jun 4 at 14:12

I am no expert but it seems to be known that $$\gamma_n>0$$.

The function that defines $$\gamma(n)$$ is

$$(-1+4z^2) \Lambda(1/2+z)=8\xi(1/2+z)$$,

where $$\xi(s)=(1/2)s(s-1)\Lambda(s)$$ is the Riemann $$\xi$$ function so that

we have

$$\gamma(n)=8\frac{n! \xi^{2n}(1/2) }{(2n)!}$$

and the fact that $$\xi^{2n}(1/2)>0$$ follow also from M. Coffey

This agrees with $$\gamma_0=8 \xi(1/2)=-\pi^{-1/4}\Gamma(1/4) \zeta(1/2)$$ noted above.

$$\gamma_n$$ is essentially the Taylor coefficient of the rotated and folded

$$\xi(1/2+i\sqrt{-x})=\sum_{n=0}^\infty \frac{\gamma_n}{n!}x^n$$

whose zeros are $$-t_n^2$$ where $$1/2+it_n$$ are the non-trivial zeros of $$\zeta(s)$$ with $$Re(t_n)>0$$.

I was interested around three years ago on the relations among Coffey's analysis, symmetric function theory (a.k.a Newton identities), and the calculus and zeros of Sheffer Appell polynomials (the Hermite polynomial families being iconic examples) and posted the related MO-Q "Derivatives of Riemann ξ and traces of zeros", where I explored some of this, and corroborated the consistency of my analysis with (and understanding of) Coffey's via explicit numerical and analytic calculations using the entire real, even function $$\Omega(t)$$ and the known Riemann zeros, extended by Gottfried Helms in the associated MSE-Q "Sums of reciprocals of powers of the imaginary part of the nontrivial zeros of the Riemann zeta function". The correspondence between my analysis and Coffey's is spot-checked in different ways in the MO-Q and in the comments to Helm's contribution to my MSE-Q. This should inspire confidence in the conclusions of Coffey's paper and similar results by Li, as it did for me, despite your expressed qualification in your comment to the accepted answer.

See articles by Bump et al. on connections between the Hermite (Appell) polynomials and the local Riemann hypothesis. I. Schur, Appell, Faber--all closely associated with sym fct. theory--were involved in related research, inspired by Jensen if I recall correctly.

The history is interesting--see the survey article I mentioned in the MSE-Q&A comments "Zeros of entire Fourier transforms" by Dimitrov and Rusev and another by Iurato.

• In addition, I was interested in application of the total Pontryagin characteristic class polynomials of oeis.org/A231846, and this was a good test case. Jun 4 at 1:24