  The theta series of a lattice has been extensively studied in the literature
and is closely related to a critical quantity widely used in the fields of
physical layer security and cryptography, known as the flatness factor, or
equivalently, the smoothing parameter of a lattice. Both fields raise the
fundamental question of determining the (globally) maximum theta series over a
particular set of volume-one lattices, namely, the stable lattices. In this
work, we present a property of unimodular lattices, a subfamily of stable
lattices, to verify that the integer lattice $\mathbb{Z}^{n}$ achieves the
largest possible value of theta series over the set of unimodular lattices.
Such a result moves towards proving a conjecture recently stated by Regev and
Stephens-Davidowitz: any unimodular lattice, except for those lattices
isomorphic to $\mathbb{Z}^{n}$, has a strictly smaller theta series than that
of $\mathbb{Z}^{n}$. Our techniques are mainly based on studying the ratio of
the theta series of a unimodular lattice to the theta series of $\mathbb{Z}^n$,
called the secrecy ratio. We relate the Regev and Stephens-Davidowitz
conjecture with another conjecture for unimodular lattices, known in the
literature as the Belfiore-Sol\'e conjecture. The latter assumes that the
secrecy ratio of any unimodular lattice has a symmetry point, which is exactly
where the global minimum of the secrecy ratio is achieved.
