Multiplicativas

Multiplicativas, in number theory, refers to multiplicative functions. A function f: N → C is called multiplicative if f(1) = 1 and f(mn) = f(m)f(n) whenever m and n are coprime. If the condition holds for all m and n without the coprime restriction, f is called completely multiplicative. The distinction matters: completely multiplicative functions satisfy f(p^k) = f(p)^k for every prime p and k ≥ 1, while multiplicative functions are determined by their values on prime powers and their behaviour on coprime factors.

Examples include the constant function f(n) = 1, which is completely multiplicative; the identity function f(n) = n, which is completely multiplicative; and f(n) = n^k for any fixed k, which is completely multiplicative. Classic arithmetic functions that are multiplicative but not completely include Euler’s totient φ(n) and the divisor-counting function τ(n) (the number of divisors). The Möbius function μ(n) and the radical function rad(n) are also multiplicative. These functions are often defined by their values on prime powers and extended to all positive integers via multiplicativity.

Applications of multiplicative functions are central in analytic number theory. They allow the decomposition of global arithmetic sums into products over primes, yielding Euler products for Dirichlet series, such as those related to the Riemann zeta function and various L-functions. The framework of multiplicative functions underlies many results about average orders, distribution of prime factors, and convolution identities, making them a foundational concept in the study of arithmetic phenomena.