A Proof of the Riemann Hypothesis Based on a New Expression of the Completed Zeta Function

The Riemann Hypothesis (RH) is proved based on a new expression of the completed zeta function ξ(s), which was obtained through pairing the conjugate zero ρρi​ and ρi‾​​ in the Hadamard product, with consideration of zero multiplicity, i.e. \( ξ(s)=ξ(0)∏ρ(1-s/ρ)=ξ(0)∏ᵢ₌₁∞(1-s/ρ_i)(1-{s}{ρ̄_i})=ξ(0)∏ᵢ₌₁∞{(}β_i^2/α_i^2+β_i^2+(s-α_i)^2/α_i^2+β_i^2{)}^{mᵢ} \), wheree \( ξ(0)=1/2 \), \( ρ_i=α_i+jβ_i \), \( ρ̄_i=α_i-jβ_i \), with \( 0<α_i<1, β_i≠ 0, 0<|β_1|≤|β_2|≤ ⋯ \), and \( m_i ≥ 1 \) is the multiplicity of \( ρ_i \)​. Then, according to the functional equation \( ξ(s)=ξ(1-s) \), we obtain \( ∏ᵢ₌₁∞{(}1+(s-α_i)^2/β_i^2{)}^{mᵢ}=∏ᵢ₌₁∞{(}1+(1-s-α_i)^2/β_i^2{)}^{mᵢ} \), which is finally equivalent to\( α_i=1/2, 0<|β_1|<|β_2|<|β_3|<⋯, i=1, 2, 3, \) Thus, we conclude that the Riemann Hypothesis is true.