# On a lattice generalisation of the logarithm and a deformation of the Dedekind eta function

## TL;DR

This paper introduces a lattice-based generalization of the logarithm and a deformation of the Dedekind eta function, revealing new extremal properties of lattice theta functions and characterizations of the natural logarithm.

## Contribution

It develops a lattice generalization of the logarithm and demonstrates their role in optimizing a deformed eta function over lattice configurations.

## Key findings

- Minimizers of lattice theta functions maximize the deformed eta function.
- The lattice-logarithm characterizes the natural logarithm through a variational problem.
- The work links lattice structures with special functions and their extremal properties.

## Abstract

We consider a deformation $E_{L,\Lambda}^{(m)}(it)$ of the Dedekind eta function depending on two $d$-dimensional simple lattices $(L,\Lambda)$ and two parameters $(m,t)\in (0,\infty)$, initially proposed by Terry Gannon. We show that the minimizers of the lattice theta function are the maximizers of $E_{L,\Lambda}^{(m)}(it)$ in the space of lattices with fixed density. The proof is based on the study of a lattice generalization of the logarithm, called lattice-logarithm, also defined by Terry Gannon. We also prove that the natural logarithm is characterized by a variational problem over a class of one-dimensional lattice-logarithm.