Pith. sign in

REVIEW 1 major objections 4 minor 163 references

$\left(p,q\right)$-adic Analysis and the Collatz Conjecture

T0 review · 1 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A single $(p,q)$-adic function is claimed to encode all periodic points of well-behaved Collatz-type maps.

desk verdict The core Correspondence Principle is real but only for integral Hydra maps; the abstract's unqualified iff is false, and the paper needs revision before I'd trust the advertised scope. read the letter →

arxiv 2412.02902 v1 pith:CKMD5ARG submitted 2024-12-03 math.GM

Maxwell Charles Siegel This is my paper · ORCID
classification math.GM MSC 11S8237P0546S10
keywords Collatzconjecturep-adicanalysisq-adicHydramapsCorrespondencePrinciplenon-archimedeanFourierTauberiantheoremquasi-integrability
topics The Collatz Conjecture
open problems The Collatz Conjecture
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that functions from the $p$-adic integers to the $q$-adic integers, where $p$ and $q$ are distinct primes, are a natural language for Collatz-type maps. For any well-behaved semi-basic Hydra map—a Collatz-type map built from finitely many affine branches chosen by $n$ modulo $p$—that fixes $0$, it constructs a unique function $\chi_H: \mathbb{Z}_p \to \mathbb{Z}_q$ whose rational integer values at rational but non-integer $p$-adic inputs are exactly the nonzero periodic points of $H$. It then develops a Fourier analysis for such functions, introducing quasi-integrability, frames, and a non-archimedean analogue of the classical Tauberian theorem, and claims that asking whether $x$ is a periodic point is essentially the same as asking whether the translates of the Fourier transform of $\chi_H(z) - x$ span a dense subspace of a non-archimedean function space. If the construction works, Collatz-type dynamics become value-distribution questions about one explicitly defined function, and the same scheme extends to maps on $\mathbb{Z}^d$.

What carries the argument

Two constructions carry the argument. The first is the numen $\chi_H$, the unique rising-continuous solution of the branch functional equations $\chi_H(pz + j) = (a_j \chi_H(z) + b_j)/d_j$ taking $\mathbb{Z}_p$ into $\mathbb{Z}_q$; it converts strings of branch choices into rational numbers $H_{\mathbf{j}}(0)$ and then interpolates them. The second is the self-map $B_p(n) = n/(1-p^{\lambda_p(n)})$, whose $p$-adic expansion repeats the digits of $n$ forever, packaging the periodic branch string of a cycle as a rational $p$-adic integer. Around these, the paper builds a $(p,q)$-adic Fourier transform on $\mathbb{Z}_p$, with frames as a formalism for series whose convergence topology varies from point to point, and quasi-integrability as a way to define integrals and Fourier transforms for such series even when the function is not continuous. The analysis culminates in non-archimedean Tauberian theorems that connect density of translate spans to periodicity.

What would settle it

Take the shortened Collatz map $T_3$, compute $\chi_{T_3}(B_2(n))$ for all $n$ up to a large bound using the explicit series from (2.110), and iterate $T_3$ directly on every integer value obtained. The Correspondence Principle predicts each returned integer is a periodic point of $T_3$; the discovery of an integer value whose direct iteration never returns to itself would refute Corollary 2.3.

Watch

Extended reading notes

Core claim

The central discovery is the Correspondence Principle. For an integral semi-basic $p$-Hydra map with $H(0)=0$, define $\chi_H$ on finite strings of branch choices by $\chi_H(\mathbf{j}) = H_{\mathbf{j}}(0)$, then interpolate to $\mathbb{Z}_p$ by the rising-continuity limit $\chi_H(z) = \lim_n \chi_H([z]_{p^n})$, which converges in $\mathbb{Z}_q$ because the multipliers $M_H(\mathbf{j})$ are $q$-adically small whenever the string contains many nonzero digits. The principle states that the set of all nonzero periodic points of $H$ in $\mathbb{Z}$ equals $\mathbb{Z} \cap \chi_H(\mathbb{Q} \cap \mathbb{Z}'_p)$, where $\mathbb{Z}'_p$ is $\mathbb{Z}_p$ minus the nonnegative integers. In particular, every cycle of length at least two contains a point of the form $\chi_H(B_p(n))$ with $n \geq 1$ and $B_p(n) = n/(1-p^{\lambda_p(n)})$, and the identity $\chi_H(B_p(n)) = \chi_H(n)/(1 - M_H(n))$ is the geometric-series engine behind this. The later Tauberian spectral theorem adds that if the span of the translates of the Fourier transform of $\chi_H(z) - x$ is dense in the space $c_0(\widehat{\mathbb{Z}}_p, \mathbb{C}_q)$ of $\mathbb{C}_q$-valued functions vanishing at infinity on the dual group of $\mathbb{Z}_p$, then $x$ is not a periodic point; the paper's formulation leaves periodicity or unbounded divergence in the non-dense case.

Load-bearing premise

The converse half of the Correspondence Principle depends on Lemma 2.6: for a semi-basic Hydra map with prime $p$, integrality and propriety are the same, meaning that applying the wrong branch to any $p$-adic integer always produces a non-integral $p$-adic number; if this failed, an integer value $\chi_H(z)$ could arise from a misapplied composition and not be a genuine periodic point.

Editorial extensions

If this is right

  • Every cycle of $H$ of length at least two contains a point expressible as $\chi_H(B_p(n))$ for some $n \geq 1$; when $H$ is integral, any $\chi_H(B_p(n))$ that is an integer is a periodic point of $H$.
  • The nonzero periodic points of an integral semi-basic Hydra map are completely described by the integer values of $\chi_H$ on $\mathbb{Q} \cap \mathbb{Z}'_p$, so cycle-finding becomes a value-distribution problem for one $(p,q)$-adic function.
  • If the span of the translates of the Fourier transform of $\chi_H(z) - x$ is dense in $c_0(\widehat{\mathbb{Z}}_p, \mathbb{C}_q)$, then $x$ is not periodic; in the non-dense case the paper's theory leaves only periodicity or an unbounded trajectory.
  • The same construction and Correspondence Principle hold for multi-dimensional Hydra maps on lattices $\mathbb{Z}^d$ and rings of algebraic integers, so the method is not specific to the classical Collatz map.
  • The paper's $(p,q)$-adic integration theory provides a new analytic toolkit for any function from $\mathbb{Z}_p$ to $\mathbb{C}_q$ built from such series, independent of the Collatz examples.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the Correspondence Principle holds in full, the weak Collatz conjecture becomes the explicit claim that no integer outside $\{1,2,4\}$ lies in the image of $\chi_{T_3}$ on $\mathbb{Q} \cap \mathbb{Z}'_2$.
  • The proved density-to-nonperiodicity direction suggests a quantitative research program: establish density of translate spans for most $x$ through $p$-adic Fourier estimates, yielding non-probabilistic non-periodicity statements that complement results asserting that almost every integer has finite stopping time.
  • The integrality-propriety step is the main technical dividing line; if non-integral Hydra maps can be conjugated to integral ones without changing periodic-point structure, the Correspondence Principle would cover classes of maps for which this paper's converse currently does not apply.
  • For number-field Collatz systems, where current knowledge is mostly heuristic, the multi-dimensional version replaces Markov-chain heuristics with exact functional equations for $\chi_H$.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The dissertation proposes a new framework, "(p,q)-adic analysis," for studying Collatz-type maps, which it calls Hydra maps. For a Hydra map H fixing 0, the author defines a function χ_H: Z_p → Z_q and proves several versions of a "Correspondence Principle": periodic points of H are related to integer values of χ_H on certain rational p-adic integers. The strongest precise form, Corollary 2.3, states that for an integral semi-basic p-Hydra map the nonzero periodic points in Z equal Z ∩ χ_H(Q ∩ Z'_p). The later chapters develop a theory of rising-continuous functions, frames, quasi-integrability, and (p,q)-adic Wiener Tauberian theorems, and apply this to the Fourier analysis of χ_H. The visible Chapter 2 proofs, including the interpolation Lemma 2.4, the decay estimate Proposition 2.12, and the geometric-series Lemma 2.5, are internally coherent at the level of detail shown.

Significance. If the formal theorem for integral maps is correct, this is a genuinely new reformulation of the periodic-point problem for a broad class of Collatz-type maps: it packages the Böhm-Sontacchi diophantine criterion into a single (p,q)-adic function and connects it to density questions in a non-archimedean function space. The construction of χ_H from the branches of H is self-contained and not circular, and the explicit estimates in Chapter 2 give the reader concrete tools. However, the advertised equivalence is stated much more broadly than what is proved. The abstract claims an iff for general Collatz-type maps, while the actual theorem requires integrality; without integrality the claimed correspondence is false. The significance of the paper is therefore contingent on carefully restricting the statement to integral semi-basic Hydra maps and revising all unqualified formulations.

major comments (1)
  1. [Abstract; §1.1.2; §2.2.3] The proof of the converse direction of Corollary 2.3 rests on Lemma 2.6, where "integral implies proper" uses the primality of p to force each d_j to be 1 or p. This is exactly the step that fails for the non-integral counterexample above. The paper should make this dependence explicit in the abstract and introduction, and should state the Correspondence Principle only for integral semi-basic p-Hydra maps, or explicitly add a discussion of the non-integral case as a separate, weaker phenomenon.
minor comments (4)
  1. [§1.1.2; Theorem 4.6] The phrase "essentially equivalent" in the abstract and introduction is stronger than what is proved: the text itself notes in a footnote that only the direction "translate-span density implies non-periodicity" is established, while the converse may leave divergent orbits. Please state this asymmetry directly in the abstract.
  2. [Preface; §2.2.3] The Preface claims a "one-to-one correspondence" between periodic points and rational integer values of χ_H, but Corollary 2.3 establishes only an equality of sets. Multiple p-adic inputs can represent the same periodic point (e.g., cyclic shifts of the branch string), so the stronger wording should be corrected.
  3. [§2.2.2, proof of Lemma 2.4] The cross-reference "By Lemma 2.67" appears to be a typo; the intended reference is Lemma 2.4. Please fix the numbering references throughout the document.
  4. [§2.2.3 opening] The subsection begins "THROUGHOUT THIS SUBSECTION, WE ASSUME H IS INTEGRAL," but Theorem 2.6(I), Corollary 2.1(I), and Corollary 2.2(I) are stated without integrality. State the standing hypotheses explicitly for each result to avoid ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Correspondence Principle is derived from the branch structure of H, not assumed as an input.

full rationale

The numen χ_H is genuinely constructed from the affine branches of H (Definition 2.14; explicit formula in Proposition 2.10), and its (p,q)-adic interpolation is proved by q_H-adic estimates (Lemma 2.4) without invoking periodic points. The Correspondence Principle (Theorem 2.6 and Corollaries 2.1–2.3) follows from the composition identity H_j(x)=M_H(j)x+χ_H(j) together with geometric-series summation; it is a derived equivalence, not a renaming of the conclusion. The converse direction for Corollary 2.3 explicitly uses the integrality hypothesis through Lemma 2.6, and the paper states "Throughout this subsection, we assume H is integral" before proving the principle, while also noting which direction does not need integrality. The abstract's unqualified iff is an overstatement for non-integral Hydra maps, but that is a scope/correctness concern rather than a circularity, since the proof does not secretly assume the periodic-point characterization. Self-references such as the author's earlier paper [143] appear only as background and are not load-bearing for the main derivation. Thus no circular step can be exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 3 invented entities

The framework is purely structural; the central claims depend on no numbers fitted to data. In the reviewed portion, free parameters are absent: the prime q is not chosen freely but defined as the smallest prime divisor of the gcd of the non-1 branch multipliers a_j (monogenicity), so it is determined by H. The key axioms are the domain restrictions that define which Hydra maps the theory covers: semi-simplicity and monogenicity for the q-adic decay that makes the interpolation converge; integrality (equivalently propriety, by Lemma 2.6) for the converse correspondence; and the unstated condition µ_0/p ≠ 1 needed in the uniqueness proof. Two background tools are also load-bearing: the universality of the geometric series, used to justify mixing real and q-adic limits, and the embedding convention for roots of unity, which makes q-adic Fourier computations well-defined.

assumptions (5)
  • domain assumption µ_0/p = a_0/d_0 must not equal 1 for the p-Hydra map H (needed to force f(0) = 0 in the uniqueness proof of the χ_H extension).
    The paper asserts this is forced by Definitions 2.1 and 2.3, but a_0 = d_0 = 1, b_0 = 0 satisfies all stated Hydra axioms with µ_0/p = 1, so the f(0) = 0 step in Lemma 2.3 needs the condition stated explicitly. The Collatz map has a_0/d_0 = 1/2, so the main application is unaffected.
  • domain assumption The Hydra map must be monogenic and semi-simple, so that a prime q_H divides all non-identity branch multipliers a_j and all d_j are coprime to q_H.
    This is the substance behind the abstract's 'appropriate choice of distinct primes p, q'. It guarantees the q_H-adic decay of M_H(j) (Proposition 2.12) that makes χ_H interpolate to Z_{q_H}; the paper also uses it to ensure branch maps are continuous on Z_{q_H} (Claim 2.1). Maps failing monogenicity are excluded from the framework.
  • domain assumption The Hydra map must be integral: each branch (a_j n + b_j)/d_j is an integer if and only if n ≡ j mod p.
    Integrality, equivalent to propriety for p prime by Lemma 2.6, is required for the converse direction of the Correspondence Principle (Theorem 2.6(II), Corollary 2.3). The paper explicitly excludes its own non-integral examples (Matthews' map M, Leigh's L1, L2) from this part of the theory.
  • standard math Geometric series universality (Fact 1.4): a series of rational terms that converges in two valued fields extending Q converges to the same rational sum in both.
    Invoked in Lemma 2.5, Remark 2.12, and the interpolation proofs to interchange real and q-adic limits when computing χ_H(B_p(n)); the author uses it explicitly to avoid Hensel's 1905 error of assuming that a p-adic convergent series of rationals equals its real value. Cited to Gouvea [36].
  • domain assumption Embedding convention for roots of unity: e^{2πi/p^n} denotes a compatible system of primitive p^n-th roots in C_q, chosen with specified digits.
    Section 1.3.4, Assumption (Embedding Convention). This makes q-adic-valued Fourier computations with complex exponentials well-defined; the paper argues the choice is immaterial because the identity in (1.60) is invariant under Gal(Q̄/Q).
invented entities (3)
  • χ_H, the numen of a Hydra map independent evidence
    purpose: A (p,q)-adic interpolation function whose integer values at rational non-integer p-adic inputs are exactly the nonzero periodic points of H, per Corollary 2.3.
    The Correspondence Principle is proved from the functional equations of χ_H and reproduces the Böhm-Sontacchi criterion (Corollary 2.2) and the T_3 cycle {1,2} (Example 2.8) as checkable special cases, giving a falsifiable handle outside the paper's framework.
  • Frames
    purpose: A bookkeeping formalism for (p,q)-adic Fourier series whose convergence topology varies from point to point; the setting in which quasi-integrability is defined (Section 3.3.3).
    A new definitional device local to this paper; it carries no empirical content and no external falsifiable handle, so it is an internal tool rather than a discovered entity.
  • Quasi-integrability
    purpose: A generalized (p,q)-adic integration theory extending Monna-Springer integration to rising-continuous functions that lack ordinary Fourier transforms (Section 3.3.5).
    A new mathematical notion whose value is internal, it enables the Fourier analysis of χ_H and the Tauberian Spectral Theorem; no independent check is provided in the reviewed portion.

how reviews work

0 comments
Cite this review

Pith. "Pith review of $\left(p,q\right)$-adic Analysis and the Collatz Conjecture." pith.science (2026). https://pith.science/paper/CKMD5ARG

@misc{pith2026241202902,
  author       = {Pith},
  title        = {Pith review of: $\left(p,q\right)$-adic Analysis and the Collatz Conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CKMD5ARG}},
  note         = {Machine review of arXiv:2412.02902}
}
abstract

What use can there be for a function from the $p$-adic numbers to the $q$-adic numbers, where $p$ and $q$ are distinct primes? The traditional answer, courtesy of the half-century old theory of non-archimedean functional analysis: not much. It turns out this judgment was premature. '$\left(p,q\right)$-adic analysis' of this sort appears to be naturally suited for studying the infamous Collatz map and similar arithmetical dynamical systems. Given such a map $H:\mathbb{Z}\rightarrow\mathbb{Z}$, one can construct a function $\chi_{H}:\mathbb{Z}_{p}\rightarrow\mathbb{Z}_{q}$ for an appropriate choice of distinct primes $p,q$ with the property that $x\in\mathbb{Z}\backslash\left\{ 0\right\} $ is a periodic point of $H$ if and only if there is a $p$-adic integer $\mathfrak{z}\in\left(\mathbb{Q}\cap\mathbb{Z}_{p}\right)\backslash\left\{ 0,1,2,\ldots\right\} $ so that $\chi_{H}\left(\mathfrak{z}\right)=x$. By generalizing Monna-Springer integration theory and establishing a $\left(p,q\right)$-adic analogue of the Wiener Tauberian Theorem, one can show that the question 'is $x\in\mathbb{Z}\backslash\left\{ 0\right\} $ a periodic point of $H$?' is essentially equivalent to 'is the span of the translates of the Fourier transform of $\chi_{H}\left(\mathfrak{z}\right)-x$ dense in an appropriate non-archimedean function space?' This presents an exciting new frontier in Collatz research, and these methods can be used to study Collatz-type dynamical systems on the lattice $\mathbb{Z}^{d}$ for any $d\geq1$.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

163 extracted references · 77 canonical work pages

  1. [1]

    A problem around Mahler functions

    Adamczewski, Boris, and Jason Bell. “A problem around Mahler functions.” (2012). <https://hal.archives-ouvertes.fr/hal-00798303/document>

    2012

  2. [2]

    Transcendental numbers in thep-adic domain

    Adams, William Wells. “Transcendental numbers in thep-adic domain”. Columbia University, 1964

    1964

  3. [3]

    Vector measures and integral operators

    Aguayo, J. “Vector measures and integral operators”. Ultrametric Functional Analysis, Con- temp. Math., vol. 384, Amer. Math. Soc., Providence, RI (2005), pp. 1-13

    2005

  4. [4]

    Non-Archimedean vector measures and integral operators

    Aguayo, J. and T. E. Gilsdorf. “Non-Archimedean vector measures and integral operators” p-Adic Functional Analysis, Lect. Notes Pure Appl. Math., vol. 222, Marcel Dekker, New York (2001), pp. 1-11

    2001

  5. [5]

    A radon nikodym theorem in the non-archimedean setting

    José Aguayo and Mirta Moraga. “A radon nikodym theorem in the non-archimedean setting.” Proyecciones (Antofagasta) 20.3 (2001): 263-279

    2001

  6. [6]

    Why is the3x + 1 Problem Hard?

    Akin, Ethan (2004), “Why is the3x + 1 Problem Hard?”, In: Chapel Hill Ergodic Theory Workshops (I. Assani, Ed.), Contemp. Math. vol 356, Amer. Math. Soc. 2004, pp. 1–20. (MR 2005f:37031)

    2004

  7. [7]

    Amice, Yvette.Les nombresp-adiques. Vol. 14. Presses universitaires de France, 1975

    1975

  8. [8]

    The p-adic ergodic theory and ap- plications

    Anashi, Vladimir Sergeevich (2014). The p-adic ergodic theory and ap- plications. <https://www.researchgate.net/publication/269571423_The_p- adic_ergodic_theory_and_applications>/

    work page arXiv 2014

Show all 163 references
  1. [9]

    On total stopping times under3x+1 iteration

    Andaloro, Paul. “On total stopping times under3x+1 iteration”, Fibonacci Quart. 38 (2000), no. 1, 73–78. MR 1738650

    2000

  2. [10]

    Density bounds for the3x + 1 problem. I. Tree- search method

    Applegate, David, and Jeffrey C. Lagarias. “Density bounds for the3x + 1 problem. I. Tree- search method.” mathematics of computation 64.209 (1995): 411-426

    1995

  3. [11]

    Density bounds for the 3x + 1 problem. II. Krasikov inequalities

    Applegate, David, and Jeffrey C. Lagarias. “Density bounds for the 3x + 1 problem. II. Krasikov inequalities.” Mathematics of computation 64.209 (1995): 427-438

    1995

  4. [12]

    The 3x + 1 semigroup

    David Applegate, Jeffrey C. Lagarias, “The 3x + 1 semigroup”, Journal of Number Theory, Volume 117, Issue 1, 2006, Pages 146-159, ISSN 0022-314X. <https://www.sciencedirect.com/science/article/pii/S0022314X05001459>

    2006

  5. [13]

    Generalized Analytic Functions

    Arens, Richard and I. M. Singer (1956). “Generalized Analytic Functions”. Transactions of the American Mathematical Society, Vol. 81, No. 2 (Mar., 1956), pp. 379-393. Published by: American Mathematical Society. <http://www.jstor.org/stable/1992923>. BIBLIOGRAPHY 448

    1956

  6. [14]

    On teaching mathematics

    Arnold, V. I., “On teaching mathematics”. Translated from the original Russian by A.V. Goryunov. Published in: Uspekhi Mat. Nauk 53 (1998), no. 1, 229-234; English transla- tion: Russian Math. Surveys 53 (1998), no. 1, 229-236. <http://www.ceremade.dauphine.fr/~ msfr/articles/...

    1998

  7. [15]

    On the sub-coordinate representation of p- adic functions

    Axelsson, Ekaterina Yurova. “On the sub-coordinate representation of p- adic functions”. Advances in Ultrametric Analysis , edited by Alain Escas- sut, et al., American Mathematical Society, 2018. ProQuest Ebook Central, <https://ebookcentral.proquest.com/lib/socal/detail.acti...

    2018

  8. [16]

    Cambridge Mathematical Library (2nd ed.), Cambridge University Press, ISBN 978-0-521-39791-9, MR 0422171

    Baker, Alan (1990).Transcendental number theory. Cambridge Mathematical Library (2nd ed.), Cambridge University Press, ISBN 978-0-521-39791-9, MR 0422171

    1990

  9. [17]

    A p-adic proof that pi is transcendental

    Baker, Matt. “A p-adic proof that pi is transcendental”.Matt Baker’s Math Blog. 20 March

  10. [18]

    Natural Boundaries and Spectral Theory

    Barry Simon and Jonathan Breuer (2011). “Natural Boundaries and Spectral Theory”

    2011

  11. [19]

    3x + 1 inverse orbit generating functions almost always have natural boundaries

    Bell, Jason P., and Jeffrey C. Lagarias. “3x + 1 inverse orbit generating functions almost always have natural boundaries.” arXiv preprint arXiv:1408.6884 (2014)

    2014 arXiv

  12. [20]

    Harmonic analysis on the p-adic numbers

    Bell, Jordan. (2016) “Harmonic analysis on the p-adic numbers”. <https://pdfs.semanticscholar.org/9d4f/f3efad867fe867331bb00402807e0856170c.pdf>

    2016

  13. [21]

    The Pontryagin duals of Q/Z and Q

    Bell, Jordan. (2015) “The Pontryagin duals of Q/Z and Q”. <https://pdfs.semanticscholar.org/bffd/643d972eab01c0b063c34ad1560791899d3a.pdf>

    2015

  14. [22]

    Visions of the Universe: A Coloring Journey Through Math’s Great Mysteries

    Alek Bellos and Edmund Harriss. Visions of the Universe: A Coloring Journey Through Math’s Great Mysteries. Experiment; Clr Csm edition. 29 Nov. 2016. ISBN-10: 1615193677

    2016

  15. [23]

    An Analytic Approach to the Collatz3n + 1Problem for Negative Start Values

    Berg, Lothar, and Gerhard Opfer. “An Analytic Approach to the Collatz3n + 1Problem for Negative Start Values.” Computational Methods and Function Theory 13.2 (2013): 225-236

    2013

  16. [24]

    The 3n+1CollatzProblemandFunctionalEquations

    Berg, Lothar, andGünterMeinardus.“The 3n+1CollatzProblemandFunctionalEquations.” (1995)

    1995

  17. [25]

    Functional equations connected with the Collatz prob- lem

    Lothar Berg and Günter Meinardus. “Functional equations connected with the Collatz prob- lem.” Results in Mathematics 25, no. 1-2 (1994): 1-12

    1994

  18. [26]

    Besikovitch, A. S. (1954)Almost Periodic Functions. Dover Publications, Inc., through special permission of Cambridge University Press

    1954

  19. [27]

    Goldie, and Jozef L

    Bingham, Nicholas H., Charles M. Goldie, and Jozef L. Teugels.Regular variation. No. 27. Cambridge university press, 1989

    1989

  20. [28]

    Borm, George Florimond.p-adic Fourier theory. Diss. [Sl: sn], 1988

    1988

  21. [29]

    Bosch, U

    S. Bosch, U. Güntzer, and R. Remmert, Non-Archimedean Analy- sis, Grundlehren der Math. Wiss. 261, Springer-Verlag, Berlin, 1984. <https://www.math.arizona.edu/~cais/scans/BGR-Non_Archimedean_Analysis.pdf>

    1984

  22. [30]

    Bump, Daniel, et al.An introduction to the Langlands program. Vol. 140. Springer Science & Business Media, 2003. BIBLIOGRAPHY 449

    2003

  23. [31]

    An algebraic fractal approach to Collatz Conjecture

    Chabrera, Víctor Martín. “An algebraic fractal approach to Collatz Conjecture”. BS thesis. Universitat Politècnica de Catalunya, 2019

    2019

  24. [32]

    Lectures on non-archimedean function theory

    Cherry, William. “Lectures on non-archimedean function theory.” arXiv preprint arXiv:0909.4509 (2009). <https://arxiv.org/pdf/0909.4509.pdf>

    2009 arXiv

  25. [33]

    (2008).Number theory: Volume II: Analytic and modern tools(Vol

    Cohen, H. (2008).Number theory: Volume II: Analytic and modern tools(Vol. 240). Springer Science and Business Media

    2008

  26. [34]

    Letter to Professor Mays

    Collatz, Lothar. “Letter to Professor Mays”. Translated from the German by Marion Meudt and John Read11

  27. [35]

    Fontaine’s rings and p-adic L-functions

    Colmez, Pierre. “Fontaine’s rings and p-adic L-functions.” Lecture notes 32 (2004): 33. <http://people.math.jussieu.fr/~ colmez/tsinghua.pdf>

    2004

  28. [36]

    Infinite Series inp-Adic Fields

    Conrad, K. “Infinite Series inp-Adic Fields”

  29. [37]

    The Local-Global Principle

    Conrad, K. “The Local-Global Principle”

  30. [38]

    Unpredictable Iterations

    Conway, John H. (1972). “Unpredictable Iterations”. Proc. 1972 Number Theory Conf., Univ. Colorado, Boulder, pp. 49–52

    1972

  31. [39]

    On the existence of cycles of given length in integer sequences likexn + 1 =xn/2 if xn even, andxn+1 = 3xn + 1 otherwise

    Corrado Böhm and Giovanna Sontacchi (1978). “On the existence of cycles of given length in integer sequences likexn + 1 =xn/2 if xn even, andxn+1 = 3xn + 1 otherwise”, Atti Ac- cad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. 64 (1978), 260–264. (MR 83h:10030)

    1978

  32. [40]

    American Mathematical Soc., 2013

    Davis, Philip J.Circulant matrices. American Mathematical Soc., 2013

    2013

  33. [41]

    CRC press, 2018

    Devaney, Robert L.An introduction to chaotic dynamical systems. CRC press, 2018

    2018

  34. [42]

    Springer, 2016

    Diagana, Toka, and François Ramaroson.Non-Archimedean operator theory. Springer, 2016

    2016

  35. [43]

    p-Adic mathematical physics: the first 30 years

    Dragovich, B., et al. “p-Adic mathematical physics: the first 30 years.” P-Adic numbers, ultrametric analysis and applications 9.2 (2017): 87-121

    2017

  36. [44]

    Adelic harmonic oscillator

    Dragovich, Branko. “Adelic harmonic oscillator.” International Journal of Modern Physics A 10.16 (1995): 2349-2365

    1995

  37. [45]

    Springer, 2015

    Ducros, Antoine, CharlesFavre, andJohannesNicaise, eds.Berkovich spaces and applications. Springer, 2015

    2015

  38. [46]

    On the rationality of the zeta function of an algebraic variety

    Dwork, Bernard. “On the rationality of the zeta function of an algebraic variety”. Amer. J. Math., 82:631–648, 1960

    1960

  39. [47]

    World Scientific, 2015

    Escassut, Alain.Value Distribution in p-adic Analysis. World Scientific, 2015

    2015

  40. [48]

    Statistical Properties of an Iterated Arithmetic Mapping

    M. R. Feix, M. Muriel and J. L. Rouet. “Statistical Properties of an Iterated Arithmetic Mapping”, J. Stat. Phys., 76 (1994), 725-741

    1994

  41. [49]

    Springer, 2014

    Federer, Herbert.Geometric measure theory. Springer, 2014

    2014

  42. [50]

    Analytic combinatorics

    Flajolet, Philippe, and Robert Sedgewick. Analytic combinatorics. Cambridge University press, 2009. 11<http://www.cecm.sfu.ca/organics/papers/lagarias/paper/goodies/ubersetzung/html/ubersetzung.html> BIBLIOGRAPHY 450

    2009

  43. [51]

    Mellin transforms and asymptotics: Harmonic sums

    Philippe Flajolet, Xavier Gourdon, and Philippe Dumas. “Mellin transforms and asymptotics: Harmonic sums”. In: Theoretical Computer Science 144 (1995), pp. 3–58

    1995

  44. [52]

    Mellin transforms and asymptotics: digital sums

    Philippe Flajolet, et al. “Mellin transforms and asymptotics: digital sums”. In: Theoretical Computer Science 123 (1994), pp. 291–314

    1994

  45. [53]

    Folland, Gerald B.Real analysis: modern techniques and their applications. Vol. 40. John Wiley and Sons, 1999

    1999

  46. [54]

    Folland, Gerald B.A course in abstract harmonic analysis. Vol. 29. CRC press, 2016

    2016

  47. [55]

    2013.Ergodic Theory of P-Adic Transformations

    Furno, Joanna. 2013.Ergodic Theory of P-Adic Transformations. PhD Thesis. University of North Carolina at Chapel Hill. <https://doi.org/10.17615/8p5w-9v68>

    2013 doi

  48. [56]

    Bounded analytic functions

    Garnett, John. Bounded analytic functions. Vol. 236. Springer Science & Business Media, 2007

    2007

  49. [57]

    Encoding and Visualization in the Collatz Conjecture

    Georgiou, George M. “Encoding and Visualization in the Collatz Conjecture.” arXiv preprint arXiv:1811.00384 (2018)

    2018 arXiv

  50. [58]

    (2011).Automorphic Representations and L-Functions for the General Linear Group: Volume 1(Vol

    Goldfeld, D., and Hundley, J. (2011).Automorphic Representations and L-Functions for the General Linear Group: Volume 1(Vol. 129). Cambridge University Press

    2011

  51. [59]

    Springer Science and Business Media, 2012

    Gouvêa, Fernando Quadros.p-adic Numbers: An Introduction. Springer Science and Business Media, 2012

    2012

  52. [60]

    Hensel’s p-adic Numbers: early history

    Gouvêa, Fernando Quadros. “Hensel’s p-adic Numbers: early history” <https://www- fourier.ujf-grenoble.fr/~panchish/Mag2009L3/GouveaHensel2.pdf>

  53. [61]

    Walsh Functions and Their Generalizations

    B. Golubov, A. Efimov, and V. Skvortsov. “Walsh Functions and Their Generalizations.” Walsh Series and Transforms. Springer, Dordrecht, 1991. 1-33

    1991

  54. [62]

    A fractal set associated with the Collatz problem

    Hashimoto, Yukihiro. “A fractal set associated with the Collatz problem.” Bull. of Aichi Univ. of Education, Natural Science 56 (2007): 1-6

    2007

  55. [63]

    Discrete dynamics of one dimensional Collatz like integral value transforma- tions

    Hassan, Sk. “Discrete dynamics of one dimensional Collatz like integral value transforma- tions.” Journal of Applied Mathematics and Computing 49.1 (2015): 91-105

    2015

  56. [64]

    Hardy, G. H. (1949).Divergent Series. Oxford University Press. Oxford: 1949

    1949

  57. [65]

    G. H. Hardy and Marcel Reisz. (1915).The general theory of Dirichlet’s series. Cambridge [Eng.] University Press, 1915. <http://hdl.handle.net/2027/mdp.39015064521282>

    1915

  58. [66]

    On a theorem of Szegő

    Helson, Henry (1955). “On a theorem of Szegő”. Proceedings of the American Mathematical Society

    1955

  59. [67]

    Ueber die Zurückführung der Divisorensysteme auf eine reducirte Form

    Hensel, Kurt. "Ueber die Zurückführung der Divisorensysteme auf eine reducirte Form." (1897): 234-250

  60. [68]

    MacGregor.Fractional Cauchy Transforms

    Hibschweiler, Rita A., and Thomas H. MacGregor.Fractional Cauchy Transforms. Chapman and Hall/CRC, 2020

    2020

  61. [69]

    Princeton Univer- sity Press, 2016

    Iwasawa, Kinkichi.Lectures onP-AdicL-Functions.(AM-74), Volume 74. Princeton Univer- sity Press, 2016. BIBLIOGRAPHY 451

    2016

  62. [70]

    Explorations of the Collatz Conjecture (modm)

    Jackson, Glenn Micah Jr., “Explorations of the Collatz Conjecture (modm)” (2014). Honors College Theses. 70. <https://digitalcommons.georgiasouthern.edu/honors-theses/70>

    2014

  63. [71]

    The3x + 1 fractal

    Joseph, L. Pe. “The3x + 1 fractal.” Computers & Graphics 28.3 (2004): 431-435

    2004

  64. [72]

    Non-archimedean function spaces and the Lebesgue dominated convergence theorem

    J. Kąkol. C. Perez-Garcia. W. Śliwa. “Non-archimedean function spaces and the Lebesgue dominated convergence theorem.” Bull. Belg. Math. Soc. Simon Stevin 19 (1) 173 - 184, march

  65. [73]

    Onp-adic vector measure spaces

    Katsaras. A. K. “Onp-adic vector measure spaces”, J. Math. Anal. Appl. ,365(2010), 342-357

    2010

  66. [74]

    Kedlaya, Kiran S.p-adic Differential Equations. Vol. 125. Cambridge University Press, 2010

    2010

  67. [75]

    Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models

    Khrennikov, Andrei. Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models. Springer, Dordrecht, 1997. 101-129

    1997

  68. [76]

    Generalized probabilities taking values in non-Archimedean fields and in topological groups

    Khrennikov, A. Yu. “Generalized probabilities taking values in non-Archimedean fields and in topological groups.”" Russian Journal of Mathematical Physics 14.2 (2007): 142-159

    2007

  69. [77]

    The measure-theoretical ap- proach top-adic probability theory

    Khrennikov, Andrei; Yamada, Shinichi; van Rooij, Arnoud. “The measure-theoretical ap- proach top-adic probability theory”. Annales Mathématiques Blaise Pascal, Tome 6 (1999) no. 1, pp. 21-32. <http://www.numdam.org/item/AMBP_1999__6_1_21_0/>

    1999

  70. [78]

    Distributional asymptotics andp-adic Tauberian and Shannon–Kotelnikov theorems

    Khrennikov, A. Yu, and V. M. Shelkovich. “Distributional asymptotics andp-adic Tauberian and Shannon–Kotelnikov theorems.” Asymptotic Analysis 46.2 (2006): 163-187

    2006

  71. [79]

    Koblitz, Neal.p-adic analysis: A short course on recent work. Vol. 46. Cambridge University Press, 1980

    1980

  72. [80]

    Koblitz, Neal.p-adic Numbers,p-adic Analysis, and Zeta-Functions. Vol. 58. Springer Science & Business Media, 2012

    2012

  73. [81]

    Non-Archimedean Duality: Algebras, Groups, and Multipliers

    Kochubei, Anatoly N. “Non-Archimedean Duality: Algebras, Groups, and Multipliers”. 2016 <https://arxiv.org/pdf/1510.06876.pdf>

    2016 arXiv

  74. [82]

    RCWA: Residue-Class-Wise Affine Groups

    Kohl, Stefan (2016). RCWA: Residue-Class-Wise Affine Groups . Version 4.0.0 <https://www.gap-system.org/Manuals/pkg/rcwa/doc/manual.pdf>

    2016

  75. [83]

    Structure Theorem for (d,g,h )-Maps

    A. V. Kontorovich and Y. G. Sinai. “Structure Theorem for (d,g,h )-Maps”, 2006 <https://arxiv.org/pdf/math/0601622.pdf>

    2006 arXiv

  76. [84]

    Stochastic Models for the 3x+1 and 5x+1 Problems

    Alex V. Kontorovich and Jeffery C. Lagarias. (2009) “Stochastic Models for the 3x+1 and 5x+1 Problems”. <https://arxiv.org/pdf/0910.1944>

    2009 arXiv

  77. [85]

    Korevaar, Jacob.Tauberian theory: a century of developments. Vol. 329. Springer Science and Business Media, 2013

    2013

  78. [86]

    How many numbers satisfy the3x + 1 conjecture?

    Krasikov, Ilia. “How many numbers satisfy the3x + 1 conjecture?” International Journal of Mathematics and Mathematical Sciences 12.4 (1989): 791-796

    1989

  79. [87]

    Tate’s thesis

    Kudla, Stephen S. “Tate’s thesis.” An introduction to the Langlands program. Birkhäuser, Boston, MA, 2004. 109-131. BIBLIOGRAPHY 452

    2004

  80. [88]

    Gibbs phenomenon for Fourier partial sums onZp

    Kyung Soo Rim. “Gibbs phenomenon for Fourier partial sums onZp”, Journal of Mathemat- ical Analysis and Applications, Volume 433, Issue 1, 2016, Pages 392-404, ISSN 0022-247X, <https://doi.org/10.1016/j.jmaa.2015.08.005>

    2016 doi

  81. [89]

    American Mathematical Soc., 2010

    Lagarias, Jeffrey C., ed.The ultimate challenge: The3x+1 problem. American Mathematical Soc., 2010

    2010

  82. [90]

    The3x+1 problem and its generalizations

    Lagarias, J.C. “The3x+1 problem and its generalizations”, Amer. Math. Monthly 92 (1985), 3–23

    1985

  83. [91]

    The set of rational cycles for the3x+1 problem

    Lagarias, Jeffrey C. “The set of rational cycles for the3x+1 problem”, Acta Arith. 56 (1990), no. 1, 33–53. MR 1067980, DOI <https://doi.org/10.4064/aa-56-1-33-53>

    1990 doi

  84. [92]

    Lang, Serge.Algebraic number theory. Vol. 110. Springer Science and Business Media, 2013

    2013

  85. [93]

    The Collatz conjecture and De Bruijn graphs

    Laarhoven, Thijs, and Benne de Weger. “The Collatz conjecture and De Bruijn graphs.” Indagationes Mathematicae 24.4 (2013): 971-983

    2013

  86. [94]

    A Markov process underlying the generalized Syracuse algorithm

    Leigh, G.M. “A Markov process underlying the generalized Syracuse algorithm”, Acta Arith. 46 (1986) 125-143

    1986

  87. [95]

    A Paley-Wiener Theorem for Locally Compact Abelian Groups

    Liepins, Gunar E. (1976). “A Paley-Wiener Theorem for Locally Compact Abelian Groups”. Transactions of the American Mathematical Society, Vol. 222 (Sep., 1976), pp. 193-210 Pub- lished by: American Mathematical Society. <http://www.jstor.org/stable/1997665>

    1976

  88. [96]

    Arithmetic properties of the solutions of a class of functional equations

    J. H. Loxton and A. J. van der Poorten (1981). “Arithmetic properties of the solutions of a class of functional equations”

    1981

  89. [97]

    Quasi-Invariant and Pseudo-Differentiable Measures with Values in Non- ArchimedeanFieldsonaNon-ArchimedeanBanachSpace

    Ludkovsky, S. V. “Quasi-Invariant and Pseudo-Differentiable Measures with Values in Non- ArchimedeanFieldsonaNon-ArchimedeanBanachSpace”.JMathSci128, 3428–3460(2005). <https://doi.org/10.1007/s10958-005-0280-2>

    2005 doi

  90. [98]

    The Laplace Transform for Locally Compact Abelian Groups

    Mackey, George W. (1948). “The Laplace Transform for Locally Compact Abelian Groups”. Proceedings of the National Academy of Sciences, Volume 34, Number 4, pp. 156-162. <http://www.pnas.org/content/34/4/156>

    1948

  91. [99]

    Introduction to p-adic numbers and their functions

    Mahler, Kurt. Introduction to p-adic numbers and their functions. No. 64. CUP Archive, 1973

    1973

  92. [100]

    An Interpolation Series for Continuous Functions of ap-adic Variable

    Mahler, Kurt. “An Interpolation Series for Continuous Functions of ap-adic Variable.” (1958): 23-34

    1958

  93. [101]

    Lacunary series

    Mandelbrojt, Szolem. “Lacunary series”. Rice Institute Pamphlet - Rice University Studies, 14, no. 4 (1927) Rice University: <https://hdl.handle.net/1911/8511>

    1927

  94. [102]

    (1972)Dirichlet Series: Principals and Methods

    Mandelbrojt, S. (1972)Dirichlet Series: Principals and Methods. D. Reidel Publishing Com- pany / Dordrecht-Holland

    1972

  95. [103]

    Gefland transform and spectral radius formulae for ultrametric Banach algebras

    Maïnetti, Nicolas. “Gefland transform and spectral radius formulae for ultrametric Banach algebras”UltrametricFunctionalAnalysis: SeventhInternationalConferenceonP-AdicFunc- tional Analysis, June 17-21, 2002, University of Nijmegen, the Netherlands. Vol. 319. Ameri- can Mathem...

    2002

  96. [104]

    A Generalized 3x + 1 Conjecture

    Matthews, K. R. “A Generalized 3x + 1 Conjecture”. <https://www.numbertheory.org/gnubc/challenge>

  97. [105]

    The generalized 3x + 1 mapping: George Leigh’s approach

    Matthews, K. R. “The generalized 3x + 1 mapping: George Leigh’s approach”. <http://www.numbertheory.org/keith/george.html>

  98. [106]

    Generalizationsofthe 3x+1problemandconnectionswithMarkovmatrices and chains

    Matthews, K.R.“Generalizationsofthe 3x+1problemandconnectionswithMarkovmatrices and chains” (2010) <http://www.numbertheory.org/PDFS/3x+1_slides.pdf>

    2010

  99. [107]

    A Markov approach to the generalized Syracuse algorithm

    Matthews, K. R., and Watts, A.M. (1985). “A Markov approach to the generalized Syracuse algorithm”. Acta Arithmetica, 45(1), 29-42

    1985

  100. [108]

    Some Analytic Aspects Concerning the Collatz Problem

    Meinardus, Günter. “Some Analytic Aspects Concerning the Collatz Problem.” (2001)

    2001

  101. [109]

    Multiplicative Systems on Ultra-Metric Spaces

    Memic, Nacima. “Multiplicative Systems on Ultra-Metric Spaces.” Mathematica Balkanica (2010)

    2010

  102. [110]

    Über Hasses Verallgemeinerung des Syracuse-Algorithmus (Kakutanis Problem)

    Möller, Herbert. “Über Hasses Verallgemeinerung des Syracuse-Algorithmus (Kakutanis Problem)”, Acta Arith. 34 (1978) 219-226

    1978

  103. [111]

    The sufficiency of arithmetic progressions for the 3x + 1 Conjecture

    Monks, Kenneth. “The sufficiency of arithmetic progressions for the 3x + 1 Conjecture.” Proceedings of the American mathematical society 134.10 (2006): 2861-2872

    2006

  104. [112]

    On the distribution of arithmetic sequences in the Collatz graph

    Monks, Keenan, et al. “On the distribution of arithmetic sequences in the Collatz graph.” <http://www.mathematicalgemstones.com/maria/slides/SlidesV4.pdf>

  105. [113]

    Integration non-Archimedienne

    Monna, A. and Springer, T. “Integration non-Archimedienne”, 1,2. Indag. Math., 25, 634-653 (1963)

    1963

  106. [114]

    Springer, 2006

    Nishioka, Kumiko.Mahler functions and transcendence. Springer, 2006

    2006

  107. [115]

    Remembering WH Schikhof

    Ochsenius, H., E. Olivos, and C. Perez-Garcia. “Remembering WH Schikhof.” P-Adic Num- bers, Ultrametric Analysis, and Applications 7.2 (2015): 81-95

    2015

  108. [116]

    Lothar Collatz

    J. J. O’Connor and E. F. Robertson. “Lothar Collatz”. <https://mathshistory. st- andrews.ac.uk/Biographies/Collatz/>, 2006. St Andrews University School of Mathematics and Statistics, Scotland

    2006

  109. [117]

    Maximum excursion and stopping time record-holders for the prob- lem: computational results

    T. Oliveira e Silva “Maximum excursion and stopping time record-holders for the prob- lem: computational results”. Math. Comp., 68 (1) (1999), pp. 371-384 (see website: <http://www.ieeta.pt/~ tos/3x+1.html> for additional computations)

    1999

  110. [118]

    An Analytic Approach to the Collatz3n+1 Problem

    Opfer, Gerhard. “An Analytic Approach to the Collatz3n+1 Problem”. Hamburger Beiträge zur Angewandten Mathematik Nr. 2011-09, May 2011

    2011

  111. [119]

    Belgrade, 2004

    Pavlović, Miroslav.Introduction to function spaces on the disk. Belgrade, 2004

    2004

  112. [120]

    de Gruyter, 2019

    Pavlović, Miroslav.Function classes on the unit disc. de Gruyter, 2019

    2019

  113. [121]

    The Collatz conjecture and the quantum mechanical harmonic oscillator

    Perelman, C. C., Carbó-Dorca, R. “The Collatz conjecture and the quantum mechanical harmonic oscillator”. J Math Chem 60, 145–160 (2022). <https://doi.org/10.1007/s10910- 021-01296-6>

    2022 doi

  114. [122]

    The Orlicz-Pettis property inp-adic analysis

    Perez-Garcia, C. and W. H. Schikhof. “The Orlicz-Pettis property inp-adic analysis”. Collect. Math., 43 (1992), 225-233. BIBLIOGRAPHY 454

    1992

  115. [123]

    A journey throughout the history of p-adic num- bers

    Perrin, Yvette. “A journey throughout the history of p-adic num- bers.” Advances in Ultrametric Analysis , edited by Alain Escassut, et al., American Mathematical Society, 2018. ProQuest Ebook Central, <https://ebookcentral.proquest.com/lib/socal/detail.action?docID=5347081>

    2018

  116. [124]

    Classical topics in complex function theory

    Remmert, Reinhold. Classical topics in complex function theory. Vol. 172. Springer Science and Business Media, 2013

    2013

  117. [125]

    Robert, A. M. (2013).A course in p-adic analysis(Vol. 198). Springer Science and Business Media. Chicago

    2013

  118. [126]

    Rogers, Keith McKenzie.Real and p-Adic Oscillatory Integrals. Diss. The University of New South Wales, 2004

    2004

  119. [127]

    A van der Corput lemma for thep-adic numbers

    Rogers, Keith M. “A van der Corput lemma for thep-adic numbers.” Proceedings of the American Mathematical Society (2005): 3525-3534

    2005

  120. [128]

    51, Marcel Dekker, New York, 1978

    van Rooij, A.C.M.Non-Archimedean functional analysis.Pure and Applied Math., vol. 51, Marcel Dekker, New York, 1978

    1978

  121. [129]

    Non-Archimedean analysis

    A. C. M. van Rooij and W. H. Schikhof. “Non-Archimedean analysis.” Nieuw Arch. Wisk. (3)19, 120–160 (1971)

    1971

  122. [130]

    Non-Archimedean integration theory

    Arnoud C. M van Rooij, and Wim H. Schikhof. “Non-Archimedean integration theory.” Inda- gationes Mathematicae (Proceedings). Vol. 72. No. 2. North-Holland, 1969

    1969

  123. [131]

    On the3x + 1 problem

    Roosendaal, E. “On the3x + 1 problem” <www.ericr.nl/wondrous>

  124. [132]

    The Cauchy Transform

    Ross, William T., Joseph A. Cima, and Alec L. Matheson. “The Cauchy Transform.” (2006)

    2006

  125. [133]

    Parity sequences of the3x + 1 map on the2-adic integers and Euclidean embedding

    Rozier, O. (2018). “Parity sequences of the3x + 1 map on the2-adic integers and Euclidean embedding”. arXiv preprint arXiv:1805.00133

    2018 arXiv

  126. [134]

    Schikhof, W. (1985). Ultrametric Calculus: An Introduction to p-Adic Analysis (Cam- bridge Studies in Advanced Mathematics). Cambridge: Cambridge University Press. doi:10.1017/CBO9780511623844

    1985 doi

  127. [135]

    Schikhof, W. H. Non-Archimedean Harmonic Analysis, Ph.D. Thesis, pp. 1–80 (Catholic Univ. of Nijmegen, The Netherlands, 1967)

    1967

  128. [136]

    A Radon-Nikodym theorem for non-Archimedean integrals and absolutely continuous measures on groups,

    Schikhof, W. H. “A Radon-Nikodym theorem for non-Archimedean integrals and absolutely continuous measures on groups,” Indag. Math. 33, 78–85 (1971)

    1971

  129. [137]

    An Approach to the Ultrametric Moment Problem

    Schikhof, W. H. “An Approach to the Ultrametric Moment Problem”, Trudy Mat. Inst. Steklova, 2004, Volume 245, 251–256

    2004

  130. [138]

    Banach spaces over non-Archimedean valued fields,

    Schikhof, W. H. “Banach spaces over non-Archimedean valued fields,” Topology Proc. 24, 547–581 (2001)

    2001

  131. [139]

    Nonarchimedean functional analysis

    Schneider, Peter. Nonarchimedean functional analysis. Springer Science & Business Media, 2001

    2001

  132. [140]

    p-adic Fourier theory

    Schneider, Peter, and Jeremy Teitelbaum. “ p-adic Fourier theory.” arXiv preprint math/0102012 (2001). BIBLIOGRAPHY 455

    2001 arXiv

  133. [141]

    On Wiener’s lemma for locally compact abelian groups

    Schulte, Eike. “On Wiener’s lemma for locally compact abelian groups”. arXiv:2005.05212 [math.FA] <https://arxiv.org/abs/2005.05212> (2020)

    2020 arXiv

  134. [142]

    The Diophantine equationax3 +by3 +cz3 = 0

    Selmer, E. “The Diophantine equationax3 +by3 +cz3 = 0”, Acta Arithmetica 85 (1951), 203–362

    1951

  135. [143]

    Conservation of Singularities in Functional Equations Associated to Collatz- Type Dynamical Systems; or, Dreamcatchers for Hydra Maps

    Siegel, M. C. “Conservation of Singularities in Functional Equations Associated to Collatz- Type Dynamical Systems; or, Dreamcatchers for Hydra Maps” (2019) arXiv:1909.09733v3 [math.GM] <https://arxiv.org/abs/1909.09733>

    2019 arXiv

  136. [144]

    Syracuse Random Variables and the Periodic Points of Collatz-type maps

    Siegel, M. C. “Syracuse Random Variables and the Periodic Points of Collatz-type maps” (2020) <https://arxiv.org/abs/2007.15936>

    2020 arXiv

  137. [145]

    Collatz sequences in the light of graph theory

    Sultanow, Eldar, Christian Koch, and Sean Cox. “Collatz sequences in the light of graph theory.” (2020)

    2020

  138. [146]

    Taibleson, M. H. Fourier Analysis on Local Fields. Notes, vol. 15, Princeton Univ. Press (1975)

    1975

  139. [147]

    The Collatz conjecture, Littlewood-Offord theory, and powers of 2 and 3

    Tao, Terence. (2011) “The Collatz conjecture, Littlewood-Offord theory, and powers of 2 and 3” <https://terrytao.wordpress.com/2011/08/25/the-collatz-conjecture-littlewood- offord-theory-and-powers-of-2-and-3/>

    2011

  140. [148]

    245C, Notes 2: The Fourier transform

    Tao, Terence. (2009) “245C, Notes 2: The Fourier transform” <https://terrytao.wordpress.com/2009/04/06/the-fourier-transform/>

    2009

  141. [149]

    Almost all orbits of the Collatz map attain almost bounded values

    Tao, Terence. (2019) “Almost all orbits of the Collatz map attain almost bounded values.” <https://arxiv.org/abs/1909.03562>

    2019 arXiv

  142. [150]

    thesis under Emil Artin

    Tate, John (1950),Fourier analysis in number fields and Hecke’s zeta functions, Princeton University Ph.D. thesis under Emil Artin. Reprinted in Cassels, J. W. S.; Fröhlich, Albrecht, eds. (1967), Algebraic number theory, London: Academic Press, pp. 305–347, MR 0215665

    1950

  143. [151]

    A stopping time problem on the positive integers

    Terras, Riho. (1976) “A stopping time problem on the positive integers”. Acta Arithmetica. XXX

    1976

  144. [152]

    On the existence of a density

    Terras, Riho. (1979) “On the existence of a density”. Acta Arithmetica. XXXV

    1979

  145. [153]

    On the origins of p-adic analysis

    Ullrich, Peter. “On the origins of p-adic analysis.” Symposia Gaussiana, Proc. of the 2nd Gauss Symposium (München 1993), Conference A: Mathematics and Theoretical Physics, ed. M. Behara, R. Fritsch und RG Lintz, (Berlin 1995). 1995

    1993

  146. [154]

    Generalized functions over the field ofp-adic numbers

    Vladimirov, Vasilii S. “Generalized functions over the field ofp-adic numbers.” Russian Math- ematical Surveys 43.5 (1988): 19

    1988

  147. [155]

    Fourier-Vilenkin series and analogs of Besov and Sobolev classes

    Volosivets, S. S. “Fourier-Vilenkin series and analogs of Besov and Sobolev classes.” Annales Univ. Sci. Budapest, Sect. Comp. Vol. 33. 2010

    2010

  148. [156]

    A Distributional Proof ofp-Adic Wiener Tauberian Theorem and Approxi- mation by Translates of a Function

    Volosivets, S. S. “A Distributional Proof ofp-Adic Wiener Tauberian Theorem and Approxi- mation by Translates of a Function.” p-Adic Numbers, Ultrametric Analysis and Applications 13.4 (2021): 308-315

    2021

  149. [157]

    p-adic space-time and string theory

    Volovich, Igor V.. “p-adic space-time and string theory.” Theoretical and Mathematical Physics 71 (1987): 574-576. BIBLIOGRAPHY 456

    1987

  150. [158]

    Second Edition

    Washington, Lawrence C.Introduction to Cyclotomic Fields. Second Edition. Vol. 83. New York: Springer New York, 1997

    1997

  151. [159]

    Comments on Opfer’s alleged proof of the3n + 1 Conjecture

    de Weger, Benne. “Comments on Opfer’s alleged proof of the3n + 1 Conjecture”. Tech. rep. Eindhoven University of Technology, 2011

    2011

  152. [160]

    CRC press, 2005

    Wilf, Herbert S.generatingfunctionology. CRC press, 2005

    2005

  153. [161]

    1681, Springer-Verlag, Berlin, 1998

    Wirsching, Günther J.The dynamical system generated by the3n+1 function, Lecture Notes in Mathematics, vol. 1681, Springer-Verlag, Berlin, 1998. MR 1612686

    1998

  154. [2012]

    https://doi.org/10.36045/bbms/1331153417

  155. [2015]

    <https://mattbaker.blog/2015/03/20/a-p-adic-proof-that-pi-is-transcendental/>

    2015

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.