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{{more footnotes|date=February 2013}}▼
{{About||the [[Abstract algebra|algebra]]ic meaning|Additive map}}
▲{{more footnotes|date=February 2013}}
In [[number theory]], an '''{{anchor|definition-additive_function-number_theory}}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:<ref name="Erdos1939">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. [http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1077746/ online]</ref>
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* The restriction of the [[logarithm|logarithmic function]] to '''N'''.
* The '''multiplicity''' of a prime factor ''p'' in ''n'', that is the largest exponent ''m'' for which ''p<sup>m</sup>'' divides ''n''.
* ''a''<sub>0</sub>(''n'') - the sum of primes dividing ''n'' counting multiplicity, sometimes called sopfr(''n''), the potency of ''n'' or the integer logarithm of ''n'' {{OEIS|A001414}}. For example:
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