Counting subrings of of index k

We consider the problem of determining the number of subrings of the ring Z n of a fixed index k, denoted f n ( k ) . We present a decomposition theorem for these subrings and calculate explicit expressions for the Dirichlet series generating function F n ( s ) = ∑ k = 1 ∞ f n ( k ) k − s for n ⩽ 4 and for the generating function Φ p ( x , y ) = ∑ e = 0 ∞ ∑ n = 0 ∞ f n ( p e ) x e y n / n ! modulo p. We also calculate f n ( k ) when k is not divisible by the sixth power of any prime.