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Hi:

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!

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!