An ergodic theorem of arithmetic type

Let d(n) be the divisor function. Let X-1, X-2,... be a sequence of centered iid random variables. We obtain strong laws of large numbers for the sums Sigma(N)(n=1) d(n)X-n. Let tau be any ergodic endomorphism of the torus T. We also establish a universal ergodic theorem, which in particular implies for any square integrable function f, that [GRAPHICS] for almost all x, where the random signs are independent.