Additive arithmetic function

2020 Mathematics Subject Classification: Primary: 11A25 [MSN][ZBL]

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.

Comments

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

If $f(n)$ is additive then $k^{f(n)}$, for constant $k$, is a multiplicative arithmetic function.

References

• Gérald Tenenbaum; Introduction to Analytic and Probabilistic Number Theory, ser. Cambridge studies in advanced mathematics 46 , Cambridge University Press (1995) ISBN 0-521-41261-7