A Proof of the Riemann Hypothesis via a New Expression of ξ(s)

Source abstract

The Riemann Hypothesis (RH) is proved via a new expression of the completed Riemann zeta function $\xi(s)$, obtained through pairing the conjugate zeros $\rho_i$ and $\bar{\rho}_i$ in the Hadamard product while accounting for zero multiplicities (which are uniquely determined, although their specific values remain unknown), i.e. ξ(s)=ξ(0)∏ρ(1−sρ)=ξ(0)∏i=1∞(1−sρi)(1−sρ¯i)=ξ(0)∏i=1∞(βi2αi2+βi2+(s−αi)2αi2+βi2)mi where $\xi(0)=\frac{1}{2}$, $\rho_i=\alpha_i+j\beta_i$, $\bar{\rho}_i=\alpha_i-j\beta_i$, with $0<\alpha_i<1$, $\beta_i\neq 0$, $0<|\beta_1|\leq|\beta_2|\leq \cdots$, and $m_i\geq 1$ is the multiplicity of $\rho_i/\bar\rho_i$. Then, according to the functional equation $\xi(s)=\xi(1-s)$, we have ∏i=1∞(1+(s−αi)2βi2)mi=∏i=1∞(1+(1−s−αi)2βi2)mi which, owing to the divisibility of entire functions, uniqueness of $m_i$, and the irreducibility of each real quadratic polynomial factor, is finally equivalent to αi=12,0&amp;lt;|β1|&amp;lt;|β2|&amp;lt;|β3|&amp;lt;⋯,i=1,2,3,… Thus, we conclude that the RH is true.