About the modern problems of the theory of hyperbolic zeta-functions of lattices

The article gives an expanded version of the report, the author of Made in January 30, 2015 in Moscow at an international conference, dedicated to the memory of Professor A. A. Karatsuba, held at the Mathematical Institute. Russian Academy of Sciences and Moscow State University named after M. V. Lomonosov.
The report sets out the facts from the history of the theory of hyperbolic zeta function, provides definitions and notation.
The main content of the report was focused discussion of actual problems of the theory of hyperbolic zeta function of lattices. Identified the following promising areas of current research:

• The problem of the correct order of decreasing hyperbolic zeta function in $ \alpha \to \infty $;
• The problem of existence of analytic continuation in the left half-plane $ \alpha = \sigma + it \, (\sigma \le1) $ hyperbolic zeta function of lattices $ \zeta_H (\Lambda | \alpha) $;
• Analytic continuation in the case of lattices S. M. Voronin $ \Lambda (F, q) $;
• Analytic continuation in the case of joint lattice approximations;
• Analytic continuation in the case of algebraic lattices   $ \Lambda (t, F) = t \Lambda (F) $;
• Analytic continuation in the case of an arbitrary lattice $ \Lambda$;
• The problem behavior hyperbolic zeta function of lattices   $ \zeta_H (\Lambda | \alpha) $ in the critical strip;
• The problem of values of trigonometric sums grids.

As a promising method for investigating these problems has been allocated an approach based on the study of the possibility of passing to the limit by a convergent sequence of Cartesian grids.
Bibliography: 19 titles.