Proof of Riemann Conjecture

Abstract

Using translation β = σ &#8722; 1/2 and rotation s = σ + it = 1/2 + iz , z = t &#8722; iβ , Riemann got two results: (Theorem A) the functional equation ξ ( z ) = G ( s ) ξ ( s ), where , and (Theorem B) the product expression , where z j are all roots of ξ ( z ) , including complex roots. He proposed Riemann conjecture (RC): All roots of ξ ( z ) are real. As the product expression can only be used as a tool of contradiction, we prove RC by contradiction. To avoid the zeros of ξ (1/2 + it ) , define a subset . We have basic estimate , on L ( R ) . One can construct by all real roots t j of ξ ( t ) . If ξ has no complex roots, then w ( t ) = G ( s ) ξ ( s ) for s = 1/2 + it . If the product expression has a complex root z '= t ' &#8722; iα , where 0 < a ≤ 1/2 , R ' = | z ′| > 10 , then ξ ( z ) has four complex roots ±( t ′ ± iα ) , and should contain fourth order factor p ( z ) , i.e . ξ ( z ) = w ( z ) p ( z ) . But p ( z ) can not be contained in ξ ( s ) , as we have on L ( R ) and p ( t ) ≥ 0.5( t / R ) 4 . As a result, we can rewrite ξ ( t ) = w ( t ) p ( t ) = G ( s ) ξ ( s ) p ( t ) on and get This contradicts the basic estimate. Therefore ξ ( z ) has no complex roots and RC holds.