Difference between revisions of "Additive arithmetic function"
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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. | ||
====Comments==== | ====Comments==== | ||
An arithmetic function is also called a number-theoretic function. | 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}} | ||
Latest revision as of 05:52, 15 April 2023
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
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