Cesàro’s Formula in an Algebraic Setting

Abstract

Let τ(n) denote the number of positive divisors of n. Cesàro’s formula says that the number of ordered pairs ≺a,b≻ of integers such that 1 ≤ a,b ≤ n and [a,b] = n is equal to τ(n2), where [a,b] is the least common multiple of a and b. We explain this result and related results in terms of a finite commutative semigroup of idempotents.