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An [[arithmetic function]] of one argument that satisfies the following conditions for two relatively prime integers $m,n$

$$ f(mn) = f(m) + f(n) \ . $$

An additive arithmetic function is said to be '''strongly additive''' if $f(p^a) = f(p)$ for all prime numbers $p$ and all positive integers $a \ge 1$. An additive arithmetic function is said to be '''completely additive''' if the condition $f(mn) = f(m) + f(n)$ is also satisfied for relatively non-coprime integers $m,n$ as well; in such a case $f(p^a) = a f(p)$.

Examples. The function $\Omega(n)$, which is the number of all prime divisors of the number $n$ (multiple prime divisors being counted according to their multiplicity), is an additive arithmetic function; the function $\omega(n)$, which is the number of distinct prime divisors of the number $n$, is strongly additive; and the function $\log m$ is completely additive.


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An arithmetic function is also called a number-theoretic function.