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 the multiplicity of zeros, i.e.ξ(s)=ξ(0)∏ρ(1−sρ)=ξ(0)∏i=1∞(1−sρi)(1−sρi‾)=ξ(0)∏i=1∞[βi2αi2+βi2+(s−αi)2αi2+βi2]miwheree ξ(0) = 1/2, ρi = αi + jβi, ρ¯i = αi − jβi, with 0 < αi < 1, βi ̸= 0, 0 < |β1| ≤ |β2| ≤ · · ·, and mi ≥ 1 is the multiplicity of ρi​. Then, according to the functional equation ξ(s)=ξ(1−s)ξ(s)=ξ(1−s), we obtain ∏i=1∞[1+(s−αi)2βi2]mi=∏i=1∞[1+(1−s−αi)2βi2]miwhich is finally equivalent toαi=12,0<∣β1∣<∣β2∣<∣β3∣<⋯ ,i=1,2,3,… Thus, we conclude that the Riemann Hypothesis is true.