Complex continuation of the associated zeta function

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let \[ \mathcal A=\{(a,b,c,d)\in\mathbb Z_{\ge0}^{4}:ad-bc=1\},\qquad f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}. \] Every such $f$ is positive, so its complex power means $f^s=\exp(s\log f)$ with the real logarithm. Define \[ Z(s)=\sum_{(a,b,c,d)\in\mathcal A} f(a,b,c,d)^s. \] The original published question cites a convergence baseline $s>1/2$. The authors' later arXiv version corrects this and records convergence for real $s>2/3$ and divergence for $s\le2/3$. The same positive-term estimate gives absolute and locally uniform convergence for $\operatorname{Re}s>2/3$, which is the initial holomorphic domain used here. [S3]

Statement recorded in the supplied source.

Extend the function $Z(s)$ to complex values of $s$: determine an analytic continuation beyond its initial half-plane of convergence and the complex domains on which such a continuation exists. [S2]

Short English statement

Continue the lattice sum of complex powers of triangle-inequality defects beyond its convergent half-plane.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9000003 (AMR-089-0003). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Nikita Kalinin and Mikhail Shkolnikov, The number $\pi$ and summation by $SL(2,\mathbb Z)$, Arnold Mathematical Journal (2018) §4, Questions, pp. 5--6. https://amj.math.stonybrook.edu/html-articles/Files-2015-2024/17-75/index.html.
• S3 Nikita Kalinin and Mikhail Shkolnikov, Tropical formulae for summation over a part of $SL(2,\mathbb Z)$, arXiv:1711.02089 (updated version) §1.6, Petrov's comment and Remark 7, pp. 6--7. https://arxiv.org/pdf/1711.02089.