Martingale central limit theorem for random multiplicative functions

Let be a Steinhaus or a Rademacher random multiplicative function. For a wide class of multiplicative functions f we prove that the sum ₍ ₗ (n) f (n), normalised to have mean square 1, has a limiting distribution. The limiting distribution we find is a Gaussian times the square-root of the total mass of a random measure associated with f. Our result applies to dᵦ, the z-th divisor function, as long as z is strictly between 0 and 12. Other examples of admissible f-s include any multiplicative indicator function with the property that f (p) =1 holds for a set of primes of density strictly between 0 and 12.