Alternative and arithmetic proofs of the pi identities

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 $\mathcal A=\{(a,b,c,d)\in\mathbb Z_{\ge0}^{4}:ad-bc=1\}$ and put \[ f(a,b,c,d)=\sqrt{a^2+b^2}+\sqrt{c^2+d^2}-\sqrt{(a+c)^2+(b+d)^2}. \] The two identities in question are \[ \sum_{\mathcal A}f(a,b,c,d)=2,\qquad \sum_{\mathcal A}f(a,b,c,d)^2=2-\frac\pi2. \] The comparison identity is the Basel formula $\sum_{n=1}^{\infty}n^{-2}=\pi^2/6$. In the source its methods include infinite-product coefficient comparisons and complex residues. The Gaussian integers are $\mathbb Z[i]=\{m+ni:m,n\in\mathbb Z\}$, where $i^2=-1$. The residue of a meromorphic function $g$ at $z_0$ is the coefficient of $(z-z_0)^{-1}$ in its Laurent series.

Statement recorded in the supplied source.

Find other proofs of both displayed lattice-sum identities using methods of proof of the Basel formula. Can the individual summand $f(a,b,c,d)$ be interpreted as a residue of an appropriate complex function at $(a+b)+(c+d)i$, or otherwise related arithmetically to Gaussian integers? [S2]

Short English statement

Give Basel-style proofs of both lattice identities, and find a residue or Gaussian-integer interpretation of their summand.

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 9000004 (AMR-089-0004). 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.