The question below relates to the paper ["Jensen Polynomials for the Riemann Zeta Function and Other Sequences" of Griffin, Ono, Rolen and Zagier](https://uva.theopenscholar.com/files/ken-ono/files/1902572116.full__8.pdf). 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!)



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


<https://reader.elsevier.com/reader/sd/pii/S0377042703007970?token=37B8EA85406B8E6AEBAB149B69DFF83D0CC656CAB585072D5D90E9EB60E948ED71C>


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 $Re(t\_n)>0$.


