Proof of Riemann Conjecture Based on Contradiction between Xi-Function and Its Product Expression

Abstract

Riemann proved three results: analytically continue ζ ( s ) over the whole complex plane s = σ + it with a pole s =1 ; (Theorem A) functional equation ξ ( t ) = G ( s 0 ) ζ ( s 0 ) , s 0 =1/2 + it and (Theorem B) product expression ξ 1 ( t ) by all roots of ξ ( t ). He stated Riemann conjecture (RC): All roots of ξ ( t ) are real . We find a mistake of Riemann: he used the same notation ξ ( t ) in two theorems. Theorem B must contain complex roots; it conflicts with RC. Thus theorem B can only be used by contradiction. Our research can be completed on s 0 =1/2 + it . Using all real roots r k and (true) complex roots z j = t j + ia j of ξ ( z ), define product expressions w ( t ) , w (0) = ξ (0) and Q ( t ) > 0 , Q (0) =1 respectively, so ξ 1 ( t ) = w ( t ) Q ( t ). Define infinite point-set L ( ω ) = { t : t ≥10 and | ζ ( s 0 )| = ω } for small ω > 0 . If ξ (t) has complex roots, then ω = ωQ ( t ) on L ( ω ) . Finally in a large interval of the first module | z 1 |>>1 , we can find many points t ∈ L ( ω ) to make Q ( t ) < 1/2 . This contraction proves RC. In addition, Riemann hypothesis (RH) ζ for also holds, but it cannot be proved by ζ .