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# Additive arithmetic functions

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*Not to be confused with additive functions, a term with a different meaning used in algebra.*

In number theory, **additive arithmetic functions** are arithmetic functions such that

where means is coprime to . Obviously, must be 0. An example is , the number of distinct prime factors of .

Erdős proved^{[1]} that if a function is additive and increasing then there is some such that for ^{[2]} In particular, if the function is integer-valued then it is uniformly zero.

## See also

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

## Notes

- ↑ 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 .