A multiplicative function is an arithmetic function f with f(1)=1 and the property f(mn)=f(m)f(n) whenever m and n are coprime. It is determined by its values at prime powers.

Principle

Multiplicativity reduces global behavior to local data at prime powers: specifying f(p^k) for all primes p and k ≥ 0 determines f on all positive integers via unique factorization.

Demonstration

Euler's totient function φ(n) is multiplicative: φ(1)=1 and for a prime power φ(p^k)=p^k-p^{k-1}; for coprime m and n, φ(mn)=φ(m)φ(n).

Misapplication

Assuming f(mn)=f(m)f(n) for all m,n without the coprimality condition; such an assumption confuses multiplicative with completely multiplicative functions and can lead to incorrect Euler products.

Consequence

Multiplicative functions admit Euler product factorizations of their Dirichlet series when coefficients are multiplicative, and their study often reduces to understanding prime-power values and local-to-global principles.

Reversal

The reversal is a non-multiplicative arithmetic function or a completely multiplicative function (which strengthens the relation to all pairs, not only coprime ones); additive functions contrast by replacing multiplication with addition on coprime arguments.

Boundary

Defined on positive integers with the coprimality condition crucial; does not force behavior for non-coprime arguments beyond the multiplicative extension from prime powers; distinction from completely multiplicative must be maintained.

Semantic Tension

Frequently confused with 'completely multiplicative' (f(mn)=f(m)f(n) for all m,n) and with homomorphisms from the multiplicative monoid of N (which is not a group), so precise hypotheses are essential.

Synthesis

A multiplicative function is an arithmetic map fixed by values on prime powers that multiplies across coprime integers, enabling Euler products and prime-factorization based analysis in multiplicative number theory.