Additive arithmetic function
An arithmetic function of one argument that satisfies the following conditions for two relatively prime integers
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An additive arithmetic function is said to be strongly additive if for all prime numbers
and all positive integers
. An additive arithmetic function is said to be completely additive if the condition
is satisfied for relatively non-prime integers
as well; in such a case
.
Examples. The function , which is the number of all prime divisors of the number
(multiple divisors are counted according to their multiplicity), is an additive arithmetic function; the function
, which is the number of different prime divisors of the number
, is strongly additive; and the function
is completely additive.
Comments
An arithmetic function is also called a number-theoretic function.
Additive arithmetic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Additive_arithmetic_function&oldid=17552