Weighted divisor sums and Bessel function series, V

Let r2(n)r2(n) denote the number of representations of nn as a sum of two squares. Finding the precise order of magnitude for the error term in the asymptotic formula for ∑n≤xr2(n)∑n≤xr2(n) is known as the circle problem  . Next, let d(n)d(n) denote the number of positive divisors of nn. Determining the exact order of magnitude of the error term associated with the asymptotic formula for ∑n≤xd(n)∑n≤xd(n) is the divisor problem. In his lost notebook, Ramanujan states without proof two identities that are associated with these two famous unsolved problems. It is natural to ask if identities exist for certain weighted sums, called Riesz sums, that generalize Ramanujan’s identities. In this paper, we establish a Riesz sum identity that generalizes Ramanujan’s identity linked to the divisor problem.