In Half-Plane $\text{Re}(s)\le1$, Riemann's $\zeta(s)$ is Convergent and the Dirichlet Series of $\zeta(s)$ is Divergent, Violating the Law of Non-Contradiction
Year: 2018
Type: preprint
Abstract: Propositional logics and Three-Valued Logics (3VL) are useful tools foranalyzing the Riemann Hypothesis (RH). Riemann's expression of $\zeta(s)$
claims to be convergent throughout half-plane $\text{Re}(s)\le1$ (except at
$s=1$). Given that the Dirichlet series expression of $\zeta(s)$ is proven to
be divergent there, the Law of Non-Contradiction (LNC) holds that $\zeta(s)$
cannot also be convergent there. So $\zeta(s)$ is exclusively divergent
throughout half-plane $\text{Re}(s)\le1$, and has no zeros. So $\zeta(1)\ne0$,
resolving the BSD conjecture in favor of finiteness, and 2D Yang-Mills theory
is invalid, due to assuming that Riemann's $\zeta(s)$ is Also, the derivation of Riemann's $\zeta(s)$ is invalid, due to Riemann's
improper use of Cauchy's integral theorem to equate the branch cut of
$f(x)=\log(-x)$ to a Hankel contour. The branch cut is non-holomorphic, and the
Hankel contour is open (or closed at infinity, thereby encircling
non-holomorphic points). Both the branch cut and Hankel contour contradict
prerequisites of Cauchy's integral theorem. In set theory, $\zeta(s)$'s zeros form an empty set, so RH (all zeros are on
the critical line) and $\overline{RH}$ (not all zeros are on the critical
line) are both vacuously true. In classical logic, $\zeta(s)$ has no zeros,
so material implication holds that RH and $\overline{RH}$ are both true, and
conjunction holds that both are false. So RH is a paradox that violates LNC and
LEM, and causes ECQ. In contrast, Frege's, {\L}ukasiewicz's, and Kleene's 3VLs reject LEM, and
hold RH to be neither true nor false. Priest's 3VL further rejects LNC and ECQ,
and holds RH to be both true and false. In Frege's logic, truth-value gaps
imply gluts, and vice versa. In Kleene's and Priest's 3VLs, the proposition if
RH, then is true, if $X$ is true (in {\L}ukasiewicz's, also if $X$ is
both), and does not cause ECQ.
Source: arXiv (Cornell University)
Author: Ayal Sharon
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2965714545