Exact estimates for integrals related to Dirichlet series (Q1964629)

The author considers integrals of the form \[ \int a(x)m^{-x} dx \] over the intervals \((0,1)\), \((1,\infty)\) and \((0,\infty)\) where \(a(x)\geq 0\). These integrals are related to the Dirichlet series \[ \sum^\infty_{m=2} a_m m^{-x}, \quad x>1,\quad a_m\geq 0. \] He extends some known results from single to double related integrals, and proves some new theorems, as well. He studies the following cases: \(a(x,y)\) is integrable on either \((0,1)^2\) or \((0,1)\times (1,\infty)\) or \((1,\infty)\times (0,1)\) or \((1,\infty)^2\), and finally on \((0,\infty)^2\). The results have the following common character: certain sums of the Dirichlet coefficients are estimated by integrals and conversely.