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!
Best Answer
In the paper:
M. W. Coffey, "Asymptotic estimation of $\xi^{(2n)}(1/2)$: On a conjecture of Farmer and Rhoades", Mathematics of Computation, {\bf 78} (2009) 1147--1154
you may find the first terms of an asymptotic expansion for $\log\xi^{(2n)}(1/2)$. From it you may get a good estimate of the coefficients $a_{2n}$.
In particular
$$\log a_{2n}=2[1-\log(4n)+\log(\log n)]n-\frac{2n}{\log n}+\frac74\log(2n)-\frac34\log(\log n)+O(1)$$