2 citations · 2 across the 10 of their papers we have counts for
11 papers
Almost-Boundedness of Collatz-Type Maps on Number Rings via Archimedean Methods
Maxwell Siegel, Rory O'Dwyer
Let be an integer , not necessarily prime, and let be a number field. In 2019, Tao showed that almost every positive integer (in the sense of logarithmic density) ha…
The Hydra Map and Numen Formalisms for Collatz-Type Problems
Maxwell C. Siegel
This paper details a generalization of the formalism presented in the author's 2024 paper, "The Collatz Conjecture and Non-Archimedean Spectral Theory - Part I - Arithmetic Dynamic…
Algebras of -Adic Distributions Induced by Pointwise Products of F-Series
Maxwell C. Siegel
Let be an integer and let be a global field. A foliated -adic F-series is a function of a -adic integer variable satisfying the functional…
-adic Analysis and the Collatz Conjecture
Maxwell Charles Siegel
What use can there be for a function from the -adic numbers to the -adic numbers, where and are distinct primes? The traditional answer, courtesy of the half-century…
Infinite Series Whose Topology of Convergence Varies From Point to Point
Maxwell C. Siegel
This paper catalogues a variety of examples concerning a type of function of a -adic integer variable defined by a formal series expression we have dubbed "-series"…
The Collatz Conjecture & Non-Archimedean Spectral Theory -- Part II -- -Adic Fourier Analysis and Wiener's Tauberian Theorem
Maxwell C. Siegel
This paper gives an overview of -adic Fourier theory - the Fourier theory of functions from the -adic numbers to the -adic numbers, where and are di…
The Collatz Conjecture & Non-Archimedean Spectral Theory -- Part I.5 -- How To Write The (Weak) Collatz Conjecture As A Contour Integral
M. C. Siegel
Let be an odd prime, and let be the Shortened map, defined by if is even and $T_{q}\left(n\right)=\…
Functional Equations Associated to Collatz-Type Maps on Integer Rings of Algebraic Number Fields
M. C. Siegel
In 1995, Meinardus & Berg presented a reformulation of the Collatz Conjecture in terms of a functional equation in a single complex variable over the open unit disk. This paper gen…
The Collatz Conjecture & Non-Archimedean Spectral Theory: Part I -- Arithmetic Dynamical Systems and Non-Archimedean Value Distribution Theory
Maxwell Charles Siegel
Let be an odd prime, and let be the Shortened map, defined by if is even and $T_{q}\left(n\right)=\…
Fourier Analysis of the Parity-Vector Parameterization of the Generalized Collatz px+1 maps
Maxwell C. Siegel
Let p be an odd prime, and consider the map H_p which sends an integer x to either x/2 or (px+1)/2 depending on whether x is even or odd. The values at x=0 of arbitrary composition…
Conservation of Singularities in Functional Equations Associated to Collatz-Type Dynamical Systems; or, Dreamcatchers for Hydra Maps
Maxwell Charles Siegel
It is known that the Collatz Conjecture (and the study of similar maps, here called "Hydra maps") can be stated in terms of solution sets of functional equations; or, equivalently,…