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Source. Theorem 4.9, p. 9, of Mikhail Neklyudov, Functional analysis approach to the Collatz conjecture, arXiv:2106.11859v9 (2022), published in Results Math. 79 (2024), no. 4, Paper No. 140, in the edition identified on the source card.

Statement

$T$ is the reduced Collatz map and $\mathcal F$ the Berg--Meinardus operator, as on the Theorem 4.4 page. The total stopping time $\sigma_\infty:\mathbb N\cup\{0\}\to\mathbb N\cup\{\infty\}$ gives, for $k\in\mathbb N$, the least $s$ with $T^s(k)=1$, and $\infty$ if there is none; $\sigma_\infty(0)=0$ (p. 6). With the convention $\lambda^\infty=0$ for $\lambda\in D$, the reduced characteristic function of Definition 4.6 (p. 8) with parameters $(0,1)$ is

$$\tilde g^\lambda_{0,1}(z)=\sum_{m\ge0}\lambda^{\sigma_\infty(m)}z^m .$$

Theorem 4.9 (p. 9). For $\lambda\in D\setminus\{0\}$,

$$\mathcal F\bigl(\tilde g^\lambda_{0,1}-1\bigr) =\frac{\tilde g^\lambda_{0,1}-1}{\lambda}+\Bigl(\lambda-\frac1\lambda\Bigr)z . \qquad(4.7)$$

The statement also writes $\tilde g^\lambda_{0,1}=g^{\lambda,1}_{0,l}$, with the letter $l$ as subscript, for the characteristic function $g^{\lambda,\beta}_{k,l}(z)=\sum_{m\ge0}\lambda^{\sigma_\infty(lm+k)}\beta^mz^{lm+k}$ of Definition 4.6 (p. 8); the equality holds for $l=1$.

Read depth. Claims checked: the identity (4.7) and its hypothesis were read on p. 9; the derivation from Example 4.8 was read through, not checked step by step.

Proof pointer

Page 9: it is Example 4.8, the case $\phi(n)=\delta_{n,1}$ of Theorem 4.5, for which $F^{\hat\phi}=\frac1{1-w^2}\sum_{n\ge1}w^{\sigma_\infty(n)}z^n$, the factor $\frac1{1-w^2}$ coming from the trivial cycle.

Dependencies

Theorem 4.5.

Bears on

No Erdős problem page cites it. It is the identity behind Corollary 4.14 and Corollary 4.17.