Conjugate Variables in Analytic Number Theory. Phase Space and Lagrangian Manifolds

For an arithmetic semigroup $(G,\partial)$, we define entropy as a function on a naturally defined continuous semigroup $\widehat G$ containing $G$. The construction is based on conditional maximization, which permits us to introduce the conjugate variables and the Lagrangian manifold corresponding to the semigroup $(G,\partial)$.