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

## TL;DR

This paper explores the connection between arithmetic-geometric mean iterations and the spectral properties of self-adjoint operators from the Heisenberg group's representation theory.

## Contribution

A new result linking Landau’s constant to the cubic arithmetic–geometric mean of 2³ and 1 is presented.

## Key findings

- Spectral bounds of self-adjoint operators follow arithmetic–geometric mean iterations from lattice structures.
- Operators resemble the identity operator as lattice density increases.
- Landau’s constant is conjectured to be half the cubic arithmetic–geometric mean of 2³ and 1.

## 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 rediscover 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 half the cubic arithmetic–geometric mean of \documentclass[12pt]{minimal}
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\begin{document}$$\root 3 \of {2}$$\end{document}23 and 1, which we believe to be a new result.