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 by pairing the conjugate zeros ρiρi​ and ρi‾ρi​​ in the Hadamard product, considering the multiplicity of zeros. Specifically, we have:ξ(s)=ξ(0)∏i(1−sρi)(1−sρi‾)ξ(s)=ξ(0)i∏​(1−ρi​s​)(1−ρi​​s​) =ξ(0)∏i(βi2αi2+βi2+(s−αi)2αi2+βi2)mi=ξ(0)i∏​(αi2​+βi2​βi2​​+αi2​+βi2​(s−αi​)2​)mi​where ξ(0)=12ξ(0)=21​, ρi=αi+jβiρi​=αi​+jβi​, ρi‾=αi−jβiρi​​=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+(s−αi)2βi2)mi=∏i(1+(1−s−αi)2βi2)mii∏​(1+βi2​(s−αi​)2​)mi​=i∏​(1+βi2​(1−s−αi​)2​)mi​Due 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,…,∞(1+βi2​(s−αi​)2​)mi​=(1+βi2​(1−s−αi​)2​)mi​,i=1,2,3,…,∞This is further equivalent to:αi=12,0<∣β1∣<∣β2∣<∣β3∣<⋯ ,i=1,2,3,…,∞αi​=21​,0<∣β1​∣<∣β2​∣<∣β3​∣<⋯,i=1,2,3,…,∞Thus, we conclude that the Riemann Hypothesis (RH) is true.