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# 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 such that

where means is coprime to . Obviously, must be 1.

For example, Euler's totient function is multiplicative: note that and thus However, as ; instead we should do . 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 such that for ^{[3]} In fact an analogous result holds for decreasing functions,^{[4]} so if a function is multiplicative and monotone then either it is for some or for

## See also

- Multiplicative arithmetic functions
- Completely multiplicative arithmetic functions
- Additive arithmetic functions
- Completely additive arithmetic functions

## 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. pp. 1–20 . - ↑ For , it is trivial, so he meant .
- ↑ Everett Howe (1986). “A new proof of Erdős's theorem on monotone multiplicative functions”.
*The American Mathematical Monthly***93**(8): pp. pp. 593–595.