A Proof of the 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 (infinite polynomial), which can be further expressed as infinite product by conjugate complex roots. Then, according to Lemma 3, Lemma 4, and Lemma 5, the functional equation \(\xi(s)=\xi(1-s)\) leads to $(s-\alpha_i)^2 = (1-s-\alpha_i)^2$ with solution \(\alpha_i= \frac{1}{2}\), where \(\alpha_i\) are the real parts of the zeros of \(\xi(s)\), i.e., \(s_i =\alpha_i\pm j\beta_i, i\in \mathbb{N}\). Thus a proof of the Riemann Hypothesis is achieved.