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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!

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    $\begingroup$ I have a short proof that a_2n greater than a_2n+2 for all n. Are you interested? $\endgroup$
    – user85015
    Commented Jan 7, 2016 at 18:56
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    $\begingroup$ Of course he is^^ $\endgroup$
    – Matthias Ludewig
    Commented Jan 7, 2016 at 22:44
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    $\begingroup$ knowing that $\xi(s)$ is an entire function, there is no doubt that $a_{2n}$ goes to $0$ quite fast. for the sign, it should be more complicated, at least than $\frac{\xi(s)}{s(s-1)} = \int_0^\infty x^{s-1}\Phi(x) dx$ which is much simpler, also even around $\Re(s) = 1/2$, but has a radius of convergence of $1/2$ at $1/2$. $\endgroup$
    – reuns
    Commented Jan 8, 2016 at 7:50
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    $\begingroup$ I have a pdf file of the proof that a_2n > a_2n+1. I can send you a copy of this proof if you are still interested. Send request to [email protected] Unfortunately, MathOverflow does not input pdf files. I do not have time to covert the file to MathJax format. $\endgroup$
    – user88789
    Commented Mar 10, 2016 at 19:51

2 Answers 2

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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

$$\log a_{2n}=2[1-\log(4n)+\log(\log n)]n-\frac{2n}{\log n}+\frac74\log(2n)-\frac34\log(\log n)+O(1)$$

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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

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  • $\begingroup$ There is no 'h' in Zagier $\endgroup$
    – Stopple
    Commented Jan 10, 2020 at 17:34

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