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

作　　者：Chuanmiao Chen Chuanmiao Chen(School of Mathematics and Statistics, Central South University, Changsha, China;College of Mathematics and Statistics, Hunan Normal University, Changsha, China)

机构地区：[1]School of Mathematics and Statistics, Central South University, Changsha, China [2]College of Mathematics and Statistics, Hunan Normal University, Changsha, China

出　　处：《Advances in Pure Mathematics》2023年第7期463-472,共10页理论数学进展（英文）

摘　　要：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(s₀)ζ (s₀), s₀ =1/2 + it and (Theorem B) product expression ξ₁(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 s₀ =1/2 + it. Using all real roots r_kand (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 ξ₁(t) = w(t)Q(t). Define infinite point-set L(ω) = {t : t ≥10 and |ζ(s₀)| =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 |z₁|>>1, we can find many points t ∈ L(ω) to make Q(t) . This contraction proves RC. In addition, Riemann hypothesis (RH) ζ for also holds, but it cannot be proved by ζ.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(s₀)ζ (s₀), s₀ =1/2 + it and (Theorem B) product expression ξ₁(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 s₀ =1/2 + it. Using all real roots r_kand (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 ξ₁(t) = w(t)Q(t). Define infinite point-set L(ω) = {t : t ≥10 and |ζ(s₀)| =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 |z₁|>>1, we can find many points t ∈ L(ω) to make Q(t) . This contraction proves RC. In addition, Riemann hypothesis (RH) ζ for also holds, but it cannot be proved by ζ.

关 键 词：Riemann Conjecture Xi-Function Functional Equation Product Expression Multiplicative Group CONTRADICTION 

分 类 号：O17[理学—数学]