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additive function
In number theory, an additive function is an arithmetic function ''f''(''n'') of the positive integer ''n'' such that whenever ''a'' and ''b'' are coprime, the function of the product is the sum of the functions:〔Erdös, P., and M. Kac. On the Gaussian Law of Errors in the Theory of Additive Functions. Proc Natl Acad Sci USA. 1939 April; 25(4): 206–207. (online )〕
:''f''(''ab'') = ''f''(''a'') + ''f''(''b'').
== Completely additive ==
An additive function ''f''(''n'') is said to be completely additive if ''f''(''ab'') = ''f''(''a'') + ''f''(''b'') holds ''for all'' positive integers ''a'' and ''b'', even when they are not co-prime. Totally additive is also used in this sense by analogy with totally multiplicative functions. If ''f'' is a completely additive function then ''f''(1) = 0.
Every completely additive function is additive, but not vice versa.
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