In [[number theory]], an '''additive function''' is an [[arithmetic function]] ''f''(''n'')ofsuch thethat positive [[integer]] ''n'' whenwhenever ''a'' and ''b'' are [[coprime]] andwe have:
:''f''(''ab'') = ''f''(''a'') + ''f''(''b'').
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 coprime.
Every completely additive function is additive, but not vice versa.
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Arithmetic functions which are completely additive are:
* AThe contractionrestriction of the [[logarithm|logarithmic function]] onto '''N'''.
* AThe function Ω(''n''), defined foras every ''n'' ≥ 2 ofthe total number of all[[prime primes,number|prime]] whichfactors devide given positive integerof ''n''., Wecounting putmultiple also Ω(1)factors =multiple 0times. Some values:
::Ω(4) = 2
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:: ...
An example of an arithmetic function which is additive but not completely additive is ω(''n''), defined as the total number of ''different'' [[prime number|prime]] factors of ''n''. Some values:
for every positive integer ''n'', where sum runs over all different [[prime number|primes]] that devide ''n'' and 1(''n'') is a constant function, defined by 1(''n'') = 1. The ω function tells us how many different primes devide arbitrary positive integer ''n''. Some values (compare with Ω(''n'')):
