Difference between revisions of "Additive arithmetic function"
From Encyclopedia of Mathematics
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| − | An arithmetic function of one argument that satisfies the following conditions for two relatively prime integers | + | 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 | + | 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 $f(p^a) = a f(p)$. |
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| + | 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==== | ====Comments==== | ||
An arithmetic function is also called a number-theoretic function. | An arithmetic function is also called a number-theoretic function. | ||
Revision as of 19:18, 15 August 2013
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.
How to Cite This Entry:
Additive arithmetic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Additive_arithmetic_function&oldid=17552
Additive arithmetic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Additive_arithmetic_function&oldid=17552
This article was adapted from an original article by I.P. Kubilyus (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article