Sums of divisors functions and Bessel function series

On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. These two identities are intimately connected with the classical circle and divisor problems, respectively. There are three possible interpretations for the double series of these identities. The first identity has been proved under all three interpretations, and the second under two of them. Furthermore, several analogues of them were established, and they were extended to Riesz sum identities as well. In this paper, we provide analogous Riesz sum identities for the weighted sums of divisors functions, and in particular two of them yield a generalization of the Riesz sum identity for r₆(n), where r₆(n) denotes the number of representations of n as a sum of six squares.