Multiplicative arithmetic functions

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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

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

[0] Outside of number theory, the term multiplicative function usually means "completely" multiplicative function, and the domain is not restricted to the positive integers.

[1] Paul Erdős (1946). "On the distribution function of additive functions". Annals of Mathematics 47 (2): pp. 1-20.

[2] For $n = 1$ , it is trivial, so he meant $n > 1$ .

[3] Everett Howe (1986). "A new proof of Erdős's theorem on monotone multiplicative functions". The American Mathematical Monthly 93 (8): pp. 593-595.

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Multiplicative arithmetic functions

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