over the unit interval (0, 1) or the interval (1, ∞) or the half-line (0, ∞), wherea(x)≥0 and is integrable on the interval in question. These integrals are related to the Dirichlet series
, where the numbersam≥0. We survey certain known results in a new formulation in order to reveal the difference in behavior between the functions
which are integrable on either (0, 1) or (1, ∞). Their proofs can be read out from the existing literature.
Second, we extend these results from single to double related integrals, while making distinction among the functionsa(x, y) which are integrable on either (0, 1)2 or (0, 1)×(1, ∞) or (1, ∞)×(0, 1) or (1, ∞)2. The case wherea(x, y) is integrable on (0, ∞)2 is also included.