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Conjugate variables in analytic number theory. Phase space and Lagrangian manifolds

Mathematical notes. 2016. Vol. 100. No. 3. P. 421-428.
Maslov V. P., Nazaikinskii V.

For an arithmetic semigroup (G), we define entropy as a function on a naturally defined continuous semigroup Ĝ 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).