Any two nonzero arithmetic
functions have a greatest common divisor relative to the Dirichlet product, but the
known proofs of this fact are nonconstructive. In a restricted setting, this paper
develops a method for obtaining specific formulas for the greatest common divisor. It
is conjectured that formulas of this type hold more generally. The method is
based on properties of a certain derivative-like operator on the Dirichlet
algebra of arithmetic functions. The resulting “differential calculus” is used to
construct polynomial equations satisfied by the greatest common divisor of
two arithmetic functions. Then the Euclidean algorithm is applied to these
polynomials.
