11 Literature inputs

The following inputs are stated in Lean and assumed by the results that use them. Poisson summation is recorded with the Fourier analysis above.

Cited theorem 1 Santens

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The integral-point counting theorem [ 28 , Theorem 8.7 ] , with its measure normalizations and auxiliary results, is used in the rank-two permutation-Picard setting. The Lean input is a general computational corollary for smooth complete Galois fans, integral subalgebras, and compatible metrics. Its hypotheses include the geometric positivity conditions and the conditions on integral points; its constant is the dual-cone volume times the normalized local integrals. Appendix B verifies those hypotheses and evaluates the constant for this surface. The linked proposition gives the full quantified input.

Cited theorem 2 Chandrasekharan–Narasimhan

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For \(E=\mathbb {Q}(\sqrt{-3})\) and \(\xi =\zeta _E\),

\[ \frac1T\int _1^T|\xi (1/2+it)|^2\, dt=O((\log T)^2)\qquad (T\to \infty ). \]

This is [ 7 , Theorem 4, (80) ] , with \([E:\mathbb {Q}]=2\) and \(\sigma =1/2\).

Cited theorem 3 Kühleitner–Nowak

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Let \(a_K(n)\) count integral ideals of norm \(n^3\) in a fixed quadratic field \(K\). If \(F\) is analytic at \(1\) and agrees with \((s-1)^2\sum _{n\geq 1}a_K(n)n^{-s}\) to its right, then

\[ \sum _{n\leq X}a_K(n)=F(1)X\log X+(F^\prime (1)-F(1))X+O\bigl(\sqrt X(\log X)^3(\log \log X)^2\bigr). \]

This is the residue form of [ 21 , Corollary 2 ] . The removal of the factor at \(3\) and the comparison with the original count are proved in Lean.

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