A Proof of the Riemann Hypothesis Based on a New Expression of the Completed Zeta Function
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The Riemann Hypothesis (RH) is proved based on a new expression of the completed zeta function ξ(s)ξ(s), which was obtained through pairing the conjugate zeros ρiρi and ρˉiρˉi in the Hadamard product, with consideration of the multiplicity of zeros, i.e., ξ(s)=ξ(0)∏ρ(1−sρ)=ξ(0)∏ρi(1−sρi)(1−sρˉi)=ξ(0)∏i(βi2αi2+βi2⋅s−αis+αi)miwhere ξ(0)=12ξ(0)=21, ρi=αi+jβiρi=αi+jβi, ρˉi=αi−jβiρˉi=αi−jβi, with 0<αi<10<αi<1, βi≠0βi=0, 0<∣β1∣<∣β2∣<⋯0<∣β1∣<∣β2∣<⋯, and mi≥1mi≥1 is the multiplicity of ρiρi. Then, according to the functional equation ξ(s)=ξ(1−s)ξ(s)=ξ(1−s), we have ∏i=1∞(1+(s−αi)2βi2)mi=∏i=1∞(1+(1−s−αi)2βi2)miDue to the divisibility contained in the above equation and the uniqueness of mimi, each polynomial factor can only divide (and thus equal) the corresponding factor on the opposite side of the equation. Therefore, we obtain (1+(s−αi)2βi2)mi=(1+(1−s−αi)2βi2)mi,i=1,2,3,…which is further equivalent to αi=12,0<∣β1∣<∣β2∣<∣β3∣<⋯ ,i=1,2,3,…Thus, we conclude that the RH is true.