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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010620/a0106201.png" />
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{{TEX|done}}{{MSC|11A25}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010620/a0106202.png" /></td> </tr></table>
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An [[arithmetic function]] of one argument that satisfies the following conditions for two relatively prime integers $m,n$
  
An additive arithmetic function is said to be strongly additive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010620/a0106203.png" /> for all prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010620/a0106204.png" /> and all positive integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010620/a0106205.png" />. An additive arithmetic function is said to be completely additive if the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010620/a0106206.png" /> is satisfied for relatively non-prime integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010620/a0106207.png" /> as well; in such a case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010620/a0106208.png" />.
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$$ f(mn) = f(m) + f(n) \ . $$
  
Examples. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010620/a0106209.png" />, which is the number of all prime divisors of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010620/a01062010.png" /> (multiple divisors are counted according to their multiplicity), is an additive arithmetic function; the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010620/a01062011.png" />, which is the number of different prime divisors of the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010620/a01062012.png" />, is strongly additive; and the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a010/a010620/a01062013.png" /> is completely additive.
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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.
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If $f(n)$ is additive then $k^{f(n)}$, for constant $k$, is a [[multiplicative arithmetic function]].
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====References====
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* 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
This article was adapted from an original article by I.P. Kubilyus (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article