Propositional Logic Applied to Three Contradictory Definitions of the Zeta Function, and to Conditionally Convergent Series

Abstract

The paper discusses the following contradictions:

(1) Contradictory definitions of the Zeta function. These three definitions of the Zeta function contradict each other: the Dirichlet series, Riemann’s Zeta function, and a third definition that is conditionally convergent throughout the critical strip and divergent throughout half-plane Re(s)≤0. If the Dirichlet series definition and one of the other two definitions are both true, then there is a contradiction regarding convergence/divergence of Zeta in the critical strip. If Riemann’s Zeta function and the third definition are both true, then there is a contradiction regarding convergence/divergence of Zeta in half-plane Re(s)≤0. And if only the Dirichlet series definition is true, than all theories that falsely assume that the other definitions are true are rendered logically unsound.

(2) The Hankel contour’s contradiction of the preconditions of Cauchy’s integral theorem. The derivation of the Riemann Zeta function uses both of these,but the contradiction between the two renders the Riemann Zeta function invalid.

(3) Conditionally convergent series. According to the Riemann series theorem, any conditionally convergent series can be rearranged to be divergent. This contradicts the associative and commutative properties of addition. It also means that all conditionally convergent series are paradoxes, and that any argument that uses a conditionally convergent series (e.g. the third definition of the Zeta function, and the definition of the Euler-Mascheroni constant) violates the Law of Non-Contradiction (LNC) in logic, and triggers Ex Contradictione Quodlibet (ECQ), also called the "Principle of Explosion".

Keywords: Propositional Logic, Foundations of Math, Law of Non-Contradiction, Paradox, Explosion, Riemann Zeta Function, Dirichlet Series, L-Functions