2021/11/10 by Maxwell C. Siegel, Siegel, M. C. · 1 citation
Mathematics · #Benford’s Law and Fraud Detection #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2111.07883
openalex publication_date 2021/11/10 · openalex created_date 2021/11/22 · openalex updated_date 2026/07/28
Let q be an odd prime, and let Tq:ℤ→ℤ be the Shortened qx+1 map, defined by Tq(n)=n/2 if n is even and Tq(n)=(qn+1)/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 T3 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 (p,q)-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 χq of Tq 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 Tq in terms of a contour integral via an application of Perron's Formula to a Dirichlet series generated by χq and the function Mq 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 \textrmRe(s)→-∞ seems to preclude direct evaluation of the contour integrals via residues, but asymptotic results may still be achievable.