Continuous extensions of Ramanujan-expandable arithmetic functions (Matthew Fox, Perimeter Institute for Theoretical Physics and Chai Karamchedu, Sandia National Labs)

We describe a natural way to continuously extend arithmetic functions that admit a Ramanujan expansion and derive the conditions under which such an extension exists. In particular, we show that the absolute convergence of a Ramanujan expansion does not guarantee the convergence of its real variable generalization. We take the divisor function as a case study, and consider how to continuously extend it to the reals.