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

Source abstract

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 ρˉiρˉ​i​ 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−jβiρˉ​i​=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.