Number-Theoretic Functions via Convolution Rings

We've all met and befriended at least one of the 'number-theoretic functions' i-, o-, (p: for a positive integer n, r(n) = the number of divisors of n, o-(n) = the sum of the divisors of n, (p(n) = the number of integers k (1 < k < n) that are relatively prime to n. Here 'divisor' means positive integral divisor; and to say that k is relatively prime to n-usually written (k, n) = 1-means that 1 is the only common divisor of k and n. For example,