Exact estimates for integrals involving Dirichlet series with nonnegative coefficients

Abstract:

We consider the Dirichlet series \[ \sum ^\infty _{k=2} a_k k^{-1-x} =: f(x), \quad x>0, \] with coefficients $a_k \ge 0$ for all $k$. Among others, we prove exact estimates of certain weighted $L^p$-norms of $f$ on the unit interval $(0,1)$ for any $0<p<\infty$, 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 involving the former one by relying on exact estimates for integrals involving the latter one. As a by-product, we obtain an analogue of the Cauchy-Hadamard criterion of (absolute) convergence of the more general Dirichlet series \[ \sum ^\infty _{k=1} c_k k^{-z}, \quad z:= x+iy, \] with complex coefficients $c_k$.