Analytic Continuation of $\zeta(s)$ Violates the Law of Non-Contradiction (LNC)

Abstract

The Dirichlet series of $\zeta(s)$ was long ago proven to be divergent throughout half-plane $\text{Re}(s)\le1$. If also Riemann's proposition is true, that there exists an "expression" of $\zeta(s)$ that is convergent at all $s$ (except at $s=1$), then $\zeta(s)$ is both divergent and convergent throughout half-plane $\text{Re}(s)\le1$ (except at $s=1$). This result violates all three of Aristotle's "Laws of Thought": the Law of Identity (LOI), the Law of the Excluded Middle (LEM), and the Law of Non-Contradition (LNC). In classical and intuitionistic logics, the violation of LNC also triggers the "Principle of Explosion" / \textit{Ex Contradictione Quodlibet} (ECQ). In addition, the Hankel contour used in Riemann's analytic continuation of $\zeta(s)$ violates Cauchy's integral theorem, providing another proof of the invalidity of Riemann's $\zeta(s)$. Riemann's $\zeta(s)$ is one of the $L$-functions, which are all invalid due to analytic continuation. This result renders unsound all theorems (e.g. Modularity, Fermat's last) and conjectures (e.g. BSD, Tate, Hodge, Yang-Mills) that assume that an $L$-function (e.g. Riemann's $\zeta(s)$) is valid. We also show that the Riemann Hypothesis (RH) is not "non-trivially true" in classical logic, intuitionistic logic, or three-valued logics (3VLs) that assign a third truth-value to paradoxes (Bochvar's 3VL, Priest's $LP$).