On the number of necklaces whose co-periods divide a given integer

Using Ramanujan sums, which generalize the Möbius function and the Euler totient function, we enumerate q -ary (fixed) necklaces over the color set { a 1 , … , a q } with n i beads of color a i for i = 1 , … , q and co-periods dividing a fixed non-negative integer v . (The co-period of a necklace is its length divided by its period.) We also provide Witt-type infinite products and Lambert-type multiple infinite series for these quantities. Furthermore, we provide regions in C q where our infinite products and series converge absolutely.