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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


  1. Paul Erdős (1946). “On the distribution function of additive functions”. Annals of Mathematics 47 (2): pp. pp. 1–20. 
  2. For , it is trivial, so he meant .