On the number of lattice points of linear comparison solutions in rectangular areas

In the theory of the hyperbolic zeta function of lattices, a significant role is played by the Bakhvalov theorem, in which the magnitude of the zeta function of the lattice of linear comparison solutions is estimated through the hyperbolic lattice parameter.
In N. M. Korobov's 1963 monograph, this theorem is proved by a method different from the original work of N. S. Bakhvalov. In this method, the central role is played by the lemma about the number of linear comparison solutions in a rectangular area.
The paper gives new estimates of the number of lattice points of linear comparison solutions in rectangular regions. This allows us to prove the strengthened Bakhvalov theorem on the evaluation of the hyperbolic zeta function of the lattice of solutions of linear comparison.
The difference between the theorem on the number of lattice points of linear comparison solutions in rectangular regions and the corresponding Korobov lemma is that instead of one estimate through the ratio of the volume of a rectangular region to a hyperbolic parameter, two more cases are added and in the first case the constant is reduced. The use of the theorem on the number of lattice points of linear comparison solutions in rectangular areas leads to the need to prove the Bakhvalov–Korobov theorem to consider various areas of application of the theorem on the number of lattice points of linear comparison solutions in rectangular areas.