Identities for trigonometric divisor sums

The goal of this paper is to prove several identities for sums of the type Sigma(n <= x)'d(n) cos(2 pi na/q), Sigma(n <= x)'nd(n) sin (2 pi na/q), Sigma(n <= x)'r(2)(n)cos (2 pi na/q), where d(n) is the number of divisors of n and r(2)(n) is the number of representations of n as a sum of two squares. Some of our identities involve weighted divisor functions, and most of our identities are new.