Exact estimates for integrals involving Dirichlet series with nonnegative coefficients

We consider the Dirichlet series [GRAPHICS] with coefficients a(k) greater than or equal to 0 for all k. Among others, we prove exact estimates of certain weighted L-p -norms of f on the unit in terval (0, 1) for any 0 < p < infinity, in terms of the coefficients a(k). Our estimation is based on the close relationship between Dirichlet series and power series. This enables us to derive exact estimates for integrals i nvolving the former one by relying on exact estimates for integrals involvi ng the latter one. As a by-product, we obtain an analogue of the Cauchy-Had amard criterion of (absolute) convergence of the more general Dirichlet ser ies [GRAPHICS] with complex coefficients c(k).