Behavior of the modulus minimum of the Dirichlet series on the system of segments

We are considering the problem of asymptotic equivalence of the logarithms of the least upper bound of the modulus of the entire sum of the Dirichlet series on a vertical line and its minimum of the modulus. In a general case we take the minimum of the modulus on some compactum that is, in a certain sense, closely approximated to a vertical segment of a fixed length. This paper provides optimal restrictions for the sequence of exponents that admit the required relation everywhere on the positive ray except the set of the finite measure probably.