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# Multiplicative arithmetic functions

Not to be confused with multiplicative functions, a term used in algebra.[1]

In number theory, multiplicative arithmetic functions are arithmetic functions $\scriptstyle a(n),\, n \,\in\, \N^+,$ such that

$m \perp n \Rightarrow a(mn) = a(m) a(n),\quad m,\, n \in \N^+, \,$

where $\scriptstyle m \,\perp\, n$ means $\scriptstyle m$ is coprime to $\scriptstyle n$. Obviously, $\scriptstyle a(1)$ must be 1.

For example, Euler's totient function $\scriptstyle \varphi(n)$ is multiplicative: note that $\scriptstyle \varphi(3) = 2$ and $\scriptstyle \varphi(7) = 6,$ thus $\scriptstyle \varphi(21) = \varphi(3) \varphi(7) = 2 \times 6 = 12.$ However, $\scriptstyle \varphi(24) \neq \phi(4) \varphi(6)$ as $\scriptstyle \gcd(4, 6) = 2$; instead we should do $\scriptstyle \varphi(24) = \varphi(3) \varphi(8)$. Functions where the identity holds even when the numbers are not coprime are called completely multiplicative.

Erdős proved[2] that if a function is multiplicative and increasing then there is some $\scriptstyle\alpha\ge0$ such that $\scriptstyle a(n)=n^\alpha$ for $\scriptstyle n\ge1.$[3] In fact an analogous result holds for decreasing functions,[4] so if a function is multiplicative and monotone then either it is $\scriptstyle a(n)=n^\alpha$ for some $\scriptstyle\alpha$ or $\scriptstyle a(n)=0$ for $\scriptstyle n>2.$

## Notes

1. Outside of number theory, the term multiplicative function usually means "completely" multiplicative function, and the domain is not restricted to the positive integers.
2. Paul Erdős (1946). "On the distribution function of additive functions". Annals of Mathematics 47 (2): pp. 1-20.
3. For n = 1, it is trivial, so he meant n > 1.
4. Everett Howe (1986). "A new proof of Erdős's theorem on monotone multiplicative functions". The American Mathematical Monthly 93 (8): pp. 593-595.