  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.
