A Proof of Riemann Hypothesis Based on MacLaurin Expansion of the Completed Zeta Function

The basic idea is to expand the completed zeta function \(\xi(s)\) in Maclaurin series. Thus, by \(\xi(s)=\xi(1-s)\), we have the following polynomial equation \begin{equation}\nonumber \begin {split} &\xi(0)+\xi^{'}(0)s+\frac{\xi^{''}(0)}{2!}s^{2}+\cdots+\frac{\xi^{(n)}(0)}{n!}s^{n}+\cdots\\ =&\xi(0)+\xi^{'}(0)(1-s)+\frac{\xi^{''}(0)}{2!}(1-s)^{2}+\cdots+\frac{\xi^{(n)}(0)}{n!}(1-s)^{n}+\cdots \end {split} \end{equation} which finally leads to \(s=1-s, s=\alpha \pm j\beta, \beta\neq 0\), then a proof of RH can be achieved.