Inversion formula for Dirichlet series and its application

A contour integration method, used to study the asymptotic of the sums of coefficients of Dirichlet series, is based on the Inversion formula. It allows you to express the sum of the coefficients in terms of the sum of the series. This approach gives effective estimates if the abscissa of absolute convergence $\sigma_a\ge 1$. In some cases, when studying arithmetical functions in generating Dirichlet series, this value is less than $1$. As a rule, in this case, the Tauberian Delange theorem, which gives only the main term of asymptotic, is applied. However, generating Dirichlet series have better analytical properties than we need for the Delange theorem application. The contour integration method allows to count on precise results, but it need the inversion formula which is effective for series with $\sigma_a< 1$.
In this paper the such inversion formula is presented and is proved to be an effective tool on examining the distribution of $d(n)$ function values in the residue classes coprim with a module. W. Narkievicz used Delange theorem to obtain the main term of the asymptotic for frequency of hits of the values of function $d(n)$ in residue classes. Application of the inversion formula allowed us to obtain more precise results.