TITLE:
Proof of Riemann Conjecture
AUTHORS:
Chuanmiao Chen
KEYWORDS:
Riemann Conjecture, Distribution of Zeros, Entire Function, Symmetry, Functional Equation, Product Expression
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.12 No.5,
May
27,
2022
ABSTRACT: Using translation β = σ −1/2 and rotation s =σ + it = 1/2 + iz, z = t − iβ, Riemann got two results: (Theorem A) the functional equation ξ(z) = G(s)ξ(s), where , and (Theorem B) the product expression , where zj 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 tj 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' − 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.