The AGM of Gauss, Ramanujan's corresponding theory, and spectral bounds of self-adjoint operators

TL;DR

This paper explores spectral bounds of self-adjoint operators linked to lattice structures, revealing connections to Ramanujan's theories, and provides new insights into Landau's constant through arithmetic-geometric mean iterations.

Contribution

It establishes novel links between spectral bounds, lattice structures, and Ramanujan's theories, including a new result on Landau's constant using cubic arithmetic-geometric means.

Findings

Spectral bounds follow arithmetic-geometric mean iterations.

Operators resemble the identity as lattice density increases.

New expression for Landau's constant as a cubic mean.

Abstract

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.