This tag is for questions relating to multiplicative functions which are arise most commonly in the field of number theory.
In number theory, a multiplicative function is an arithmetic function $~f~ \colon \mathbb N \to \mathbb C$ fulfilling $f(1)=1~~,~~f(ab)=f(a)~f(b)~~$ for $~a~$, $~b~$ coprime.
A completely multiplicative function satisfies $~f(ab)=f(a)~f(b)~$ for all values of $~a~$ and $~b~$.
Multiplicative functions arise naturally in many contexts in number theory and algebra. The Dirichlet series associated with multiplicative functions have useful product formulas, such as the formula for the Riemann zeta function.
Well-known examples of multiplicative functions:
Euler's totient functions $\varphi(n)= \text{the number of positive integers $~a\le n~~$ such that} ~~gcd(a,n)=1~.$
Divisor functions, e.g. number of divisors $d(n)$ and sum of divisors $\sigma(n)$
The Möbius function: $~\mu(n)$
- The function $e(n) = \left\lfloor\frac{1}{n}\right\rfloor == \begin{cases} ~1~ \quad \text{if}~~ n=1\\ 0 \quad \text{otherwise} \end{cases}$
- The unit function: $~\mathbf{1}(n)=1~$
- The identity function: $~I(n) = n ~$
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