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 zeros in the Hadamard product, with consideration of zero multiplicity, i. e. \ ( (s) = (0) _ (1-s) = (0) ₈=₁^ (1-sᵢ) (1-sᵢ) = (0) ₈=₁^ (ᵢ²ᵢ²+ᵢ²+ (s-ᵢ) ²ᵢ²+ᵢ²) ^m₈ \), wheree \ ( (0) =12 \), \ (ᵢ=ᵢ+jᵢ \), \ (ᵢ=ᵢ-jᵢ \), with \ (0ᵢ1, ᵢ 0, 0|₁||₂| \), and \ (mᵢ ≥ 1 \) is the multiplicity of \ (ᵢ /ᵢ \) . Then, according to the functional equation \ ( (s) = (1-s) \), we obtain \ (₈=₁^ (1+ (s-ᵢ) ²ᵢ²) ^m₈=₈=₁^ (1+ (1-s-ᵢ) ²ᵢ²) ^m₈ \), which, owing to the divisibility of entire function, uniqueness of \ (mᵢ \), and the irreducibility of each polynomial factor, is finally equivalent to \ (ᵢ=12, 0|₁||₂||₃|, i=1, 2, 3, \) Thus, we conclude that the Riemann Hypothesis is true.
Weicun Zhang (Thu,) studied this question.