The Collatz Conjecture & Non-Archimedean Spectral Theory -- Part I.5 -- How To Write The (Weak) Collatz Conjecture As A Contour Integral

Abstract

Let $q$ be an odd prime, and let $T_{q}:\mathbb{Z}\rightarrow\mathbb{Z}$ be the Shortened $qx+1$ map, defined by $T_{q}\left(n\right)=n/2$ if $n$ is even and $T_{q}\left(n\right)=\left(qn+1\right)/2$ if $n$ is odd. The study of the dynamics of these maps is infamous for its difficulty, with the characterization of the dynamics of $T_{3}$ being an alternative formulation of the famous Collatz Conjecture. This series of papers presents a new paradigm for studying such arithmetic dynamical systems by way of a neglected area of ultrametric analysis which we have termed $\left(p,q\right)$-adic analysis, the study of functions from the $p$-adics to the $q$-adics, where $p$ and $q$ are distinct primes. In this, the first-and-a-halfth paper of the series, as a first application, we show that the numen $\chi_{q}$ of $T_{q}$ can be used in conjunction with the Correspondence Principle (CP) and classic complex-analytic tools of analytic number theory to reformulate the study of periodic points of $T_{q}$ in terms of a contour integral via an application of Perron's Formula to a Dirichlet series generated by $\chi_{q}$ and the function $M_{q}$ introduced in the first paper in this series, for which we establish functional equations, which we use to derive their meromorphic continuations to the left half-plane. The hypergeometric growth of the series as $\textrm{Re}\left(s\right)\rightarrow-\infty$ seems to preclude direct evaluation of the contour integrals via residues, but asymptotic results may still be achievable.