multiplicatif

Multiplicatif, in number theory, refers to arithmetic functions that satisfy a specific multiplicativity property. An arithmetic function f: N → C is called multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever gcd(m, n) = 1. Such functions are determined by their values on prime powers p^k, since every positive integer factors uniquely into primes.

A central feature is the Euler product for Dirichlet series. If f is multiplicative, the Dirichlet series F(s) = ∑_{n≥1} f(n)/n^s factors as an Euler product F(s) = ∏_p (1 + f(p)p^{-s} + f(p^2)p^{-2s} + ...), at least in regions where the series converges absolutely. The family of multiplicative functions is closed under Dirichlet convolution: if f and g are multiplicative, then so is their convolution (f * g)(n) = ∑_{d|n} f(d)g(n/d).

Completely multiplicative functions satisfy f(mn) = f(m)f(n) for all m, n, not only when they are coprime. Examples include id(n) = n, the Liouville function λ(n), and n^k for fixed k. Many classic multiplicative functions are defined by their values on prime powers; examples include the constant 1 function, the Euler totient φ(n), the Möbius function μ(n), the divisor-counting function d(n), and the sum-of-divisors function σ(n).

Applications and significance: multiplicative functions encode how arithmetic quantities interact with prime factorization. They are foundational in analytic number theory, underpin Euler products for L-functions, and facilitate the study of average order and distribution results via Dirichlet series and related techniques.