Estimates for sums of Dirichlet series

In the article, we prove approximation theorems that allow us to estimate, with sufficient accuracy, the supremum modulus of a Dirichlet series by the maximal term of another Dirichlet series associated with the given one. Using these theorems, we establish necessary and sufficient conditions on the sequence of coefficients of a Dirichlet series, under which the most general asymptotic and global estimates from above for its supremum modulus hold.