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Multiplicative arithmetic functions
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 that if a function is multiplicative and increasing then there is some such that for  In fact an analogous result holds for decreasing functions, so if a function is multiplicative and monotone then either it is for some or for
- Multiplicative arithmetic functions
- Completely multiplicative arithmetic functions
- Additive arithmetic functions
- Completely additive arithmetic functions
- ↑ 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. http://www.renyi.hu/~p_erdos/1946-06.pdf.
- ↑ 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.