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 ρi‾ρi​​ in the Hadamard product, with consideration of the multiplicity of zeros. That is,ξ(s)=ξ(0)∏ρ(1−sρ)=ξ(0)∏i=1∞(1−sρi)(1−sρi‾)ξ(s)=ξ(0)ρ∏​(1−sρ)=ξ(0)i=1∏∞​(1−ρi​s​)(1−ρi​​s​) =ξ(0)∏i=1∞(βi2αi2+βi2+(s−αi)2αi2+βi2)mi=ξ(0)i=1∏∞​(αi2​+βi2​βi2​​+αi2​+βi2​(s−αi​)2​)mi​where ξ(0)=12ξ(0)=21​, ρi=αi+jβiρi​=αi​+jβi​, and ρi‾=αi−jβiρi​​=αi​−jβi​, with 0<αi<10<αi​<1 and βi≠0βi​=0 as real numbers. mi≥1mi​≥1 is the multiplicity of ρiρi​, and 0<∣β1∣≤∣β2∣≤…0<∣β1​∣≤∣β2​∣≤….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)mii=1∏∞​(1+βi2​(s−αi​)2​)mi​=i=1∏∞​(1+βi2​(1−s−αi​)2​)mi​Owing to the divisibility contained in the above equation and the uniqueness of mimi​, each polynomial factor can only divide (and thereby equal) the corresponding factor on the opposite side of the equation.Thus, 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 is true.