An arithmetic function maps positive integers to complex numbers. A function f is multiplicative if f(mn) = f(m)f(n) whenever gcd(m, n) = 1, and completely multiplicative if this holds for all m and n regardless of their gcd. Because every positive integer factors uniquely into prime powers, a multiplicative function is entirely determined by its values on prime powers. Key examples include Euler's totient φ(n), the divisor function σ(n), and the Möbius function μ(n). Multiplicativity enables efficient computation and is the foundation for techniques like Möbius inversion and Dirichlet series manipulation.
Verify multiplicativity by hand for small examples: compute φ(12) via φ(4)·φ(3) and confirm it matches the direct count. Then see how knowing φ(pᵏ) = pᵏ − pᵏ⁻¹ lets you compute φ for any n from its prime factorization.
Multiplicative does not mean f(mn) = f(m)f(n) for all m, n—that is completely multiplicative. The coprimality condition is essential. Also, f(1) = 1 is a consequence of multiplicativity, not an extra assumption.
An arithmetic function is any function f : ℕ → ℂ — it assigns a complex number to each positive integer. Some famous examples include the divisor-counting function d(n) (how many divisors n has), the sum-of-divisors function σ(n), and Euler's totient φ(n) (how many integers less than n are coprime to n). What makes these functions tractable is a structural property rooted in the Fundamental Theorem of Arithmetic: they are all multiplicative. A function f is multiplicative if f(mn) = f(m)f(n) whenever gcd(m, n) = 1. This condition means that the value of f at any integer is entirely determined by its values on prime powers, because the Fundamental Theorem writes every n uniquely as a product of coprime prime powers p₁^{a₁} · p₂^{a₂} · ⋯ · pₖ^{aₖ}, and multiplicativity then gives f(n) = f(p₁^{a₁}) · f(p₂^{a₂}) · ⋯ · f(pₖ^{aₖ}).
The coprimality requirement gcd(m, n) = 1 is not a technicality — it is the entire substance of the definition. Multiplicativity does not say f(mn) = f(m)f(n) for all m and n; that stronger condition defines a completely multiplicative function. The distinction matters: Euler's totient is multiplicative but not completely multiplicative. You can verify this directly: φ(4) = 2 and φ(2) = 1, but φ(4) ≠ φ(2)·φ(2) = 1, because gcd(2, 2) = 2 ≠ 1. In contrast, the identity function f(n) = n is completely multiplicative since f(mn) = mn = f(m)f(n) for all m and n, with no coprimality restriction needed.
The power of multiplicativity is computational. To compute φ(360), you do not enumerate the integers coprime to 360. Instead, factor 360 = 2³ · 3² · 5, compute φ(2³) = 4, φ(3²) = 6, φ(5) = 4, and multiply: φ(360) = 4 · 6 · 4 = 96. This works because the prime-power factors are pairwise coprime, so multiplicativity applies directly. The reduction from a problem about the full integer to a product of independent problems on prime powers is what makes multiplicative functions computationally efficient and theoretically clean.
Multiplicativity also has deep algebraic consequences. The Dirichlet convolution of two arithmetic functions, defined by (f * g)(n) = Σ_{d|n} f(d)g(n/d), preserves multiplicativity: if f and g are both multiplicative, so is f * g. This algebraic structure is the foundation for Möbius inversion — the technique that lets you recover f from its summatory function F(n) = Σ_{d|n} f(d). The Möbius function μ(n) is itself multiplicative, and its role as the inverse of the constant function 1 under Dirichlet convolution is what makes the inversion formula work. From the factorization-based viewpoint you have built here, Möbius inversion becomes a natural consequence of the ring structure of multiplicative functions under Dirichlet convolution.