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A Two-Variable Zeta Function for a Parity-Perturbed Hofstadter Q-Recursion: The Exceptional t = -1 Slice and Gaussian Boundary Layers — arXiv:2609.02412

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.02412
License (as archived): CC0-1.0
Archived: 2026-10-17T05:39:23.050Z
SHA-256 of the archived text: e374d8d73e4383148c2b18fca8e412da83795e892143553d865b29b917981aa0
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How to cite

A Two-Variable Zeta Function for a Parity-Perturbed Hofstadter Q-Recursion: The Exceptional t = -1 Slice and Gaussian Boundary Layers — arXiv:2609.02412 — arXiv (en), https://arxiv.org/abs/2609.02412 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_cefbe2d3cd49 (Metraipolis archive dc_cefbe2d3cd49, sha256 e374d8d73e4383148c2b18fca8e412da83795e892143553d865b29b917981aa0)

BibTeX

@misc{dccefbe2d3cd49,
  title = {A Two-Variable Zeta Function for a Parity-Perturbed Hofstadter Q-Recursion: The Exceptional t = -1 Slice and Gaussian Boundary Layers — arXiv:2609.02412},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.02412},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_cefbe2d3cd49; sha256 e374d8d73e4383148c2b18fca8e412da83795e892143553d865b29b917981aa0},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.02412. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

We study the parity-perturbed Hofstadter $Q$-recursion $$ \widetilde Q(1)=\widetilde Q(2)=1,\qquad \widetilde Q(n)=\widetilde Q(n-\widetilde Q(n-1)) +\widetilde Q(n-\widetilde Q(n-2))+(-1)^n, $$ and the associated two-variable Dirichlet series $$ Z_{\widetilde Q}(s,t)=\sum_{n\ge1}n^{-s}\widetilde Q(n)^{-t}. $$ The estimate $\widetilde Q(n)=n/2+O(n/\sqrt{\log n})$ gives the exact domain of absolute convergence $\operatorname{Re}(s+t)>1$. With $w=s+t$, we separate the universal term $2^t\zeta(w)$ and derive exact transport, frequency-position, and dyadic renormalization identities. The main result concerns $t=-1$. For $E(n)=2\widetilde Q(n)-n$ and $A(X)=\sum_{n\le X}E(n)$, the binary-arch clock yields $$ A(X)=X\log_2X+X\Omega\!\left(\log_2\frac{3X}{32}\right) +O\!\left(\frac{X}{\sqrt{\log X}}\right), $$ where $\Omega$ is an explicit continuous periodic function. This continues the normalized correction to $\operatorname{Re}w>0$ and yields a boundary resonance lattice: a double resonance at $w=0$ and simple resonances at $2\pi i m/\log2$. After subtracting the full-slice order-$X$ skeleton, we analyze the negative-even arch channel. Its companion-forest layers have a weak Gaussian limit, and a canonical subsequence realizes the optimal $n/\sqrt{\log n}$ pointwise scale with an explicit signed constant. The negative-arch mass satisfies $$ A_r=\frac{512}{9\sqrt{2\pi}}\frac{16^r}{\sqrt r} \left(1-\frac{13}{16r}+O(r^{-2})\right). $$ We do not claim a full-slice continuation across $\operatorname{Re}w=0$.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-03 in math.CO).
Categorie arXiv: Number Theory (math.NT) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.02412 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-17T04:59:51.803Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-03?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).