##  [Multiplicative Function](/multiplicative-function-0) 

 Definition

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

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

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

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

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

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

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

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

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.