arXiv subjects

Ayal Sharon

Publications and source records attributed to Ayal Sharon.

2 recordsLinked to original sources

arXiv2020

Quantum Electrodynamics (QED) Renormalization is a Logical Paradox, Zeta Function Regularization is Logically Invalid, and Both are Mathematically Invalid

Quantum Electrodynamics (QED) renormalizaion is a paradox. It uses the Euler-Mascheroni constant, which is defined by a conditionally convergent series. But Riemann's series theorem proves that any conditionally convergent series can be rearranged to be divergent. This contradiction (a series that is both convergent and divergent) is a paradox in "classical" logic, intuitionistic logic, and Zermelo-Fraenkel set theory, and also contradicts the commutative and associative properties of addition. Therefore QED is mathematically invalid. Zeta function regularization equates two definitions of the Zeta function at domain values where they contradict (where the Dirichlet series definition is divergent and Riemann's definition is convergent). Doing so either creates a paradox (if Riemann's definition is true), or is logically invalid (if Riemann's definition is false). We show that Riemann's definition is false, because the derivation of Riemann's definition includes a contradiction: the use of both the Hankel contour and Cauchy's integral theorem. Also, a third definition of the Zeta function is proven to be false. The Zeta function has no zeros, so the Riemann hypothesis is a paradox, due to material implication and "vacuous subjects".

math.GM↗

arXiv2019

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

The Dirichlet series of $ζ(s)$ 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 $ζ(s)$ that is convergent at all $s$ (except at $s=1$), then $ζ(s)$ 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 $ζ(s)$ violates Cauchy's integral theorem, providing another proof of the invalidity of Riemann's $ζ(s)$. Riemann's $ζ(s)$ 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 $ζ(s)$) 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 $LP$).

math.GM↗