Logarithmic means and double series of Bessel functions

In his lost notebook, Ramanujan recorded two identities involving double series of Bessel functions that are closely connected with the classical, unsolved circle and divisor problems. In a series of papers with Zaharescu, the authors proved these identities under various interpretations, as well as Riesz mean analogues. In this paper, logarithmic mean analogues, also involving double series of Bessel functions, are established. Weighted divisor sums involving characters play a central role.