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), which was obtained through paring the conjugate zeros ρi and ρ ¯ i in the Hadamard product, with consideration of the multiplicities of zeros, i.e. ξ ( s ) = ξ ( 0 ) ∏ ρ ( 1 − s ρ ) = ξ ( 0 ) ∏ i = 1 ∞ ( 1 − s ρ i ) ( 1 − s ρ ¯ i ) = ξ ( 0 ) ∏ i = 1 ∞ ( β i 2 α i 2 + β i 2 + ( s − α i ) 2 α i 2 + β i 2 ) m i where ξ ( 0 ) = 1 2 , ρ i = α i + j β i and ρ ¯ i = α i − j β i are the complex conjugate zeros of ξ(s), 0 < αi < 1 and βi ≠ 0 are real numbers, mi ≥ 1 is the multiplicity of ρi, finite and unique (see Lemma 9 and Lemma 10), 0 < |β1 | ≤ |β2| ≤ ⋯. Then, according to the functional equation ξ(s) = ξ(1 − s), we have ∏ i = 1 ∞ ( 1 + ( s − α i ) 2 β i 2 ) m i = ∏ i = 1 ∞ ( 1 + ( 1 − s − α i ) 2 β i 2 ) m i Owing to the divisibility of the infinite product of polynomial factors and the uniqueness of mi , the above equation is equivalent to (see Lemma 3 for details) ( 1 + ( s − α i ) 2 β i 2 ) m i = ( 1 + ( 1 − s − α i ) 2 β i 2 ) m i , i = 1 , 2 , 3 , … , ∞ which is further equivalent to α i = 1 2 , 0 < | β 1 | < | β 2 | < | β 3 | < ⋯ , i = 1 , 2 , 3 , … , ∞ Thus we conclude that the RH is true.