Number Theory   Analytic Continuation of the Zeta Function Violates the Law of Non-Contradiction (LNC)

Authors: Ayal Sharon

The Dirichlet series of the Zeta function was long ago proven to be divergent throughout half-plane Re(s) =< 1. If also Riemann's proposition is true, that there exists an "expression" of the Zeta function that is convergent at all values of s (except at s = 1), then the Zeta function is both divergent and convergent throughout half-plane Re(s) =< 1 (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": Ex Contradictione Quodlibet (ECQ). In addition, the Hankel contour used in Riemann's analytic continuation of the Zeta function violates Cauchy's integral theorem, providing another proof of the invalidity of analytic continuation of the Zeta function. Also, Riemann's Zeta function is one of the L-functions, which are all invalid, because they are generalizations of the invalid analytic continuation of the Zeta function. 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 function) 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).