Neklyudov 2021 functional analysis collatz

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corollary_4_14: States that the Collatz conjecture is equivalent to the eigenvalue 1 of the Berg-Meinardus operator on analytic functions having a two-dimensional eigenspace, to z/(1-z) lying in the closed span of the stopping-time polynomials, and to the iterates of the operator on z+z^2 converging to z/(1-z).

corollary_4_17: States that for every q in (0, 1/2) the sum over n >= 2 of q to the total stopping time of n is at most (2-q)q/(1-2q).

lemma_1_1: States that the sum of the monomials z^n over a cycle of the reduced Collatz map is a fixed point of the associated operator, and that every polynomial fixed point has this form up to the operator's kernel.

lemma_2_4: States that every diverging trajectory of the reduced Collatz map yields an explicit fixed point of the associated operator, the sum of the monomials along the trajectory plus a lacunary correction series.

theorem_2_2: States that if the reduced Collatz map has no nontrivial cycles then the associated operator on a quotient of the Bergman space is hypercyclic; the printed proof writes out only the case where the conjecture holds.

theorem_2_3: States that every complex number of modulus below sqrt 2 is an eigenvalue of infinite multiplicity of the Collatz operator on the Bergman-space quotient, with eigenfunctions built from lacunary series that do not come from cycles or diverging trajectories.

theorem_3_1: States that the Collatz operator, written as an operator on square-integrable functions of the circle, preserves integrals against Lebesgue measure.

theorem_4_4: States the paper's main result: the number of cycles of the reduced Collatz map on the positive integers, the trivial cycle included, is at most the index of Id minus the Collatz operator on the Hardy space of the disc.

theorem_4_5: States that for coefficients of polynomial growth of degree l the generating function of their values along Collatz orbits is analytic on the disc times the disc of radius (2/3)^l and equals the resolvent of the Berg-Meinardus operator applied to the series.

theorem_4_9: States that for every nonzero lambda in the unit disc the series of lambda^sigma(m) z^m, sigma the Collatz total stopping time, satisfies an explicit inhomogeneous eigen-equation for the Berg-Meinardus operator.

theorem_5_4: States that an explicit function FP_2, built from the lacunary series sum z^(2^p) and its compositions with powers z^(3^k), is analytic on the unit disc and is a fixed point of the Collatz operator.

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Mikhail Neklyudov, Functional analysis approach to the Collatz conjecture, arXiv:2106.11859v9 (2022), 15 pp.; published in Results Math. 79 (2024), no. 4, Paper No. 140, doi:10.1007/s00025-024-02167-7. Labels and pages here are those of arXiv v9; the published version was not compared with it.

For the reduced Collatz map $T(n)=(3n+1)/2$ ($n$ odd), $n/2$ ($n$ even) on $\mathbb Z$, the paper studies the linear operator $\mathcal T$ with $\mathcal T(z^n)=z^{T(n)}$ and its adjoint on the Hardy space $H^2(D)$, the operator $\mathcal F$ of Berg and Meinardus. Cycles of $T$ give polynomial fixed points of $\mathcal T$ (Lemma 1.1, p. 2) and a diverging trajectory would give one that is an infinite power series (Lemma 2.4, p. 4), while every $\lambda$ with $|\lambda|<\sqrt2$ is an eigenvalue of infinite multiplicity on a quotient of the Bergman space (Theorem 2.3, p. 4), so fixed points unrelated to cycles also exist. If $T$ has no nontrivial cycles, $\mathcal T$ is hypercyclic on that quotient (Theorem 2.2, p. 4). Lebesgue measure on the circle is invariant for $\mathcal T$ (Theorem 3.1, p. 5). The main result bounds the number of cycles of $T$ on $\mathbb N$, the trivial one included, by the index of $Id-\mathcal T$ on $H^2(D)$ (Theorem 4.4, p. 7), using that $\mathcal F$ is expansive (Proposition 4.2, p. 6); the paper does not compute or bound that index. Section 4 also computes the resolvent of $\mathcal F$ on series of polynomial growth (Theorem 4.5, p. 8), derives a functional equation for the generating function of total stopping times (Theorem 4.9, p. 9), restates the conjecture in three operator forms, one of them due to Berg and Meinardus (Corollary 4.14, p. 10), and bounds $\sum_{n\ge2}q^{\sigma_\infty(n)}$ for $q<\frac12$ (Corollary 4.17, p. 11). Section 5 builds a further explicit fixed point of $\mathcal T$ (Theorem 5.4, p. 13).

Read status: claims checked for the results linked below, statements read clause by clause on the printed pages of arXiv v9; no proof is checked step by step.

The copy read for this card is arXiv:2106.11859v9 (1 Jun 2022; 15 pp.). The arXiv record (https://arxiv.org/abs/2106.11859, read 2026-10-02) names the Creative Commons Attribution-NonCommercial-ShareAlike 4.0 license.

Bears on.

• #1135: the problem's map $f$ is the paper's $T$ on $\mathbb N$. Corollary 4.14 restates the problem's conjecture in three equivalent operator forms, Theorem 4.4 bounds the number of cycles by an operator index the paper does not evaluate, and Lemma 1.1, Lemma 2.4 and Theorem 2.2 translate cycles and divergent trajectories into properties of the operator. None of these decides the problem.

Results.

• Lemma 1.1 (p. 2): a cycle of $T$ gives the fixed point $\sum_iz^{n_i}$ of $\mathcal T$; polynomial fixed points have this form up to the kernel.
• Theorem 2.2 (p. 4): if $T$ has no nontrivial cycles, $\mathcal T$ is hypercyclic on $H^2_{ber}(D)/\operatorname{span}\{1,z,z^2\}$.
• Theorem 2.3 (p. 4): every $\lambda$ with $|\lambda|<\sqrt2$ is an eigenvalue of $\mathcal T$ of infinite multiplicity on that quotient.
• Lemma 2.4 (p. 4): a diverging trajectory of $T$ gives an explicit fixed point of $\mathcal T$.
• Theorem 3.1 (p. 5): Lebesgue measure on the circle is invariant for $\mathcal T$ on $L^2(\mathbb S^1,\mathbb C)$.
• Theorem 4.4 (p. 7): the number of cycles of $T$ on $\mathbb N$, trivial one included, is at most the index of $Id-\mathcal T$ on $H^2(D)$.
• Theorem 4.5 (p. 8): the resolvent $(I-w\mathcal F)^{-1}$ on series with coefficients $O((n+1)^l)$, analytic for $|w|<(2/3)^l$.
• Theorem 4.9 (p. 9): a functional equation for $\sum_m\lambda^{\sigma_\infty(m)}z^m$, $0<|\lambda|<1$.
• Corollary 4.14 (p. 10): three operator forms equivalent to the Collatz conjecture.
• Corollary 4.17 (p. 11): $\sum_{n\ge2}q^{\sigma_\infty(n)}\le(2-q)q/(1-2q)$ for $q\in(0,\frac12)$.
• Theorem 5.4 (p. 13): an explicit fixed point $FP_2$ of $\mathcal T$, analytic in the disc.

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