  We study the spectral bounds of self-adjoint operators on the Hilbert space
of square-integrable functions, arising from the representation theory of the
Heisenberg group. Interestingly, starting either with the von Neumann lattice
or the hexagonal lattice of density 2, the spectral bounds obey well-known
arithmetic-geometric mean iterations. This follows from connections to Jacobi
theta functions and Ramanujan's corresponding theories. As a consequence we
re-discover that these operators resemble the identity operator as the density
of the lattice grows. We also prove that the conjectural value of Landau's
constant is obtained as the cubic arithmetic-geometric mean of $\sqrt[3]{2}$
and 1, which we believe to be a new result.

Classical Analysis and ODEs;Functional Analysis;Metric Geometry;Number Theory