A Proof of the Riemann Hypothesis Based on MacLaurin Expansion and Hadamard Product 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 (Hadamard product) by conjugate complex roots \(\alpha_i\pm j\beta_i, i\in \mathbb{N}\). Then, according to the functional equation \(\xi(s)=\xi(1-s)\), we have \[\xi(0)\prod_{i=1}^{\infty}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}\Big{(}1+\frac{(s-\alpha_i)^2}{\beta_i^2}\Big{)} =\xi(0)\prod_{i=1}^{\infty}\frac{\beta_i^2}{\alpha_i^2+\beta_i^2}\Big{(}1+\frac{(1-s-\alpha_i)^2}{\beta_i^2}\Big{)}\] which is equivalent to $$(s-\alpha_i)^2 = (1-s-\alpha_i)^2, i \in \mathbb{N}$$ with solution \(\alpha_i= \frac{1}{2}, i\in \mathbb{N}\). Therefore, a proof of the Riemann Hypothesis can be achieved.