Evaluation of some integrals of sums involving the Möbius function

Abstract

Integrals of sums involving the Möbius 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.

Keywords:

• Möbius function
• Riemann hypothesis
• Tauberian constants
• Weak Mertens hypothesis

Acknowledgements

I would like to thank Steven Finch (Harvard University) for introducing me to the problem of computing the Tauberian constants I _k , as well as Dr. Jan van de Lune (Hallum, The Netherlands) and Professor Roger Heath-Brown FRS (Oxford University) for many instructive discussions and suggestions.

Additional information

Notes on contributors

Tadej Kotnik

Email: [email protected]