How to Prove Riemann Conjecture by Riemann’s Four Theorems

Abstract

Riemann (1859) had proved four theorems: analytic continuation ζ ( s ) , functional equation ξ ( z ) = G ( s ) ζ ( s ) ( s = 1 / 2 + i z , z = t − i ( σ − 1 / 2 ) ), product expression ξ 1 ( z ) and Riemann-Siegel formula Z ( z ) , and proposed Riemann conjecture (RC): All roots of ξ ( z ) are real. We have calculated ξ and ζ , and found that ξ ( z ) is alternative oscillation, which intuitively implies RC, and the property of ζ ( s ) is not good. Therefore Riemann’s direction is correct, but he used the same notation ξ ( t ) = ξ 1 ( t ) to confuse two concepts. So the product expression only can be used in contraction. We find that if ξ has complex roots, then its structure is destroyed, so RC holds. In our proof, using Riemann’s four theorems is sufficient, needn’t cite other results. Hilbert (1900) proposed Riemann hypothesis (RH): The non-trivial roots of ζ have real part 1/2. Of course, RH also holds, but can not be proved directly by ζ ( s ) .