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

Source abstract

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 $2^tζ(w)$ 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 $$A(X)=X\log_2X+XΩ\!\left(\log_2\frac{3X}{32}\right) +O\!\left(\frac{X}{\sqrt{\log X}}\right),$$ where $Ω$ 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 $2πi m/\log2$. 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 $$A_r=\frac{512}{9\sqrt{2π}}\frac{16^r}{\sqrt r} \left(1-\frac{13}{16r}+O(r^{-2})\right).$$ We do not claim a full-slice continuation across $\operatorname{Re}w=0$.