TITLE: References on Taylor series expansion of Riemann xi function
QUESTION [9 upvotes]: I am looking for the references on Taylor series expansion of Riemann xi function at $\frac{1}{2}$.
$$ \xi (s)=\sum_0^{\infty}a_{2n}(s-\frac{1}{2})^{2n}$$
where
$$a_{2n}=4\int_1^{\infty}\frac{d[x^{3/2}\psi'(x)]}{dx}\frac{(\frac{1}{2}ln(x))^{2n}}{(2n)!}x^{-1/4}dx$$
and
$$\psi(x)=\sum_{m=1}^{\infty}e^{-m^2\pi x}=\frac{1}{2}[\theta_3(0,e^{-\pi x})-1]$$
Specifically I would like to know how fast $a_{2n}$ goes to zero.
Has anyone proved that 
$$a_0>a_2>a_4>...>a_{2n}>...>a_{\infty}=0$$ 
Thanks a lot!

REPLY [3 votes]: An accurate asymptotic estimate for the Taylor coefficients was obtained via the saddle point method in
Griffin+Ono+Rolen+Zagier, Jensen polynomials for the Riemann zeta function and other sequences - arXiv 1902.07321 www.pnas.org/cgi/doi/10.1073/pnas.1902572116
A short verification of the asymptotic formula was posted to the Pari/GP users mailing list, with references
Learning with GP: Griffin,Ono,Rolen,Zagier asymptotic formula for xi(s)
http://pari.math.u-bordeaux.fr/archives/pari-users-1908/msg00022.html
J. Gélinas