Evaluation of some integrals of sums involving the Mobius function

Integrals of sums involving the Mobius function appear in a variety of problems. In this paper, a divergent integral related to several important properties of the Riemann zeta function is evaluated computationally. The order of magnitude of this integral appears to be compatible with the Riemann hypothesis, and furthermore the value of the multiplicative constant involved seems to be the smallest possible. In addition, eleven convergent integrals representing Tauberian constants that characterize the relations between certain summation methods are evaluated computationally to five or more digits of precision.