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An [[Arithmetic function|arithmetic function]] of one argument, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654301.png" />, satisfying the condition
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An [[Arithmetic function|arithmetic function]] of one argument, $f(m)$, satisfying the condition
  
<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/m/m065/m065430/m0654302.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
+
$$
 +
f(mn) = f(m) f(n)
 +
$$
  
for any pair of coprime integers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654303.png" />. It is usually assumed that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654304.png" /> is not identically zero (which is equivalent to the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654305.png" />). A multiplicative arithmetic function is called strongly multiplicative if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654306.png" /> for all prime numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654307.png" /> and all natural numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654308.png" />. If (*) holds for any two numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m0654309.png" />, and not just for coprime numbers, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543010.png" /> is called totally multiplicative; in this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543011.png" />.
+
for any pair of coprime integers $m,n$. It is usually assumed that $f$ is not identically zero (which is equivalent to the condition $f(1)=1$). A multiplicative arithmetic function is called '''strongly multiplicative''' if $f(p^a) = f(p)$ for all prime numbers $p$ and all natural numbers $a$. If (*) holds for any two numbers $m,n$, and not just for coprime numbers, then $f$ is called '''totally multiplicative'''; in this case $f(p^a) = f(p)^a$.
  
Examples of multiplicative arithmetic functions. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543012.png" />, the number of natural divisors of a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543013.png" />; the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543014.png" />, the sum of the natural divisors of the natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543015.png" />; the [[Euler function|Euler function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543016.png" />; and the [[Möbius function|Möbius function]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543017.png" />. The function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543018.png" /> is a strongly-multiplicative arithmetic function, a power function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543019.png" /> is a totally-multiplicative arithmetic function.
+
Examples of multiplicative arithmetic functions. The function $\tau(m)$, the number of natural divisors of a natural number $m$; the function $\sigma(m)$, the sum of the natural divisors of the natural number $m$; the [[Euler function|Euler function]] $\phi(m)$; and the [[Möbius function|Möbius function]] $\mu(m)$. The function $\phi(m)/m$ is a strongly multiplicative arithmetic function, a power function $m^s$ is a totally multiplicative arithmetic function.
  
  
  
 
====Comments====
 
====Comments====
The convolution product
+
The [[Dirichlet convolution]] product
  
<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/m/m065/m065430/m06543020.png" /></td> </tr></table>
+
$$
 +
(f*g)(n) = \sum_{d\vert n} f(d) g(n/d)
 +
$$
  
yields a [[Group|group]] structure on the multiplicative functions. The unit element is given by the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543021.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543023.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543024.png" />. Another standard multiplicative function is the constant function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543025.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543026.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543027.png" />) and its inverse <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543028.png" />, the [[Möbius function|Möbius function]]. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543029.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543030.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543031.png" />, and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543033.png" />.
+
yields a [[Group|group]] structure on the multiplicative functions. The unit element is given by the function $e$, where $e(1)=1$ and $e(m) = 0$ for all $m > 1$. Another standard multiplicative function is the constant function $\zeta(n)$ with $\zeta(m) = 1$ for all $m$ and its inverse $\mu$, the [[Möbius function|Möbius function]]. Note that $\phi = \mu * N_1$, where $N_1(n) = n$ for all $n$, and that $\tau = \zeta * \zeta$, $\sigma = \zeta * N_1$.
  
Formally, the [[Dirichlet series|Dirichlet series]] of a multiplicative function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543034.png" /> has an [[Euler product|Euler product]]:
+
Formally, the [[Dirichlet series|Dirichlet series]] of a multiplicative function $f$ has an [[Euler product|Euler product]]:
  
<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/m/m065/m065430/m06543035.png" /></td> </tr></table>
+
$$
 +
\sum_{n=1}^\infty f(n) n^{-s} = \prod_p \left({1 + f(p) p^{-s} + f(p^2) p^{-2s} + \cdots }\right) \ ,
 +
$$
  
whose form simplifies considerably if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065430/m06543036.png" /> is strongly or totally multiplicative.
+
whose form simplifies considerably if $f$ is strongly or totally multiplicative.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,  E.M. Wright,  "An introduction to the theory of numbers" , Clarendon Press  (1960)  pp. Chapts. XVI-XVII</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  G.H. Hardy,  E.M. Wright,  "An introduction to the theory of numbers" , Clarendon Press  (1960)  pp. Chapts. XVI-XVII</TD></TR></table>

Revision as of 19:02, 15 August 2013

An arithmetic function of one argument, $f(m)$, satisfying the condition

$$ f(mn) = f(m) f(n) $$

for any pair of coprime integers $m,n$. It is usually assumed that $f$ is not identically zero (which is equivalent to the condition $f(1)=1$). A multiplicative arithmetic function is called strongly multiplicative if $f(p^a) = f(p)$ for all prime numbers $p$ and all natural numbers $a$. If (*) holds for any two numbers $m,n$, and not just for coprime numbers, then $f$ is called totally multiplicative; in this case $f(p^a) = f(p)^a$.

Examples of multiplicative arithmetic functions. The function $\tau(m)$, the number of natural divisors of a natural number $m$; the function $\sigma(m)$, the sum of the natural divisors of the natural number $m$; the Euler function $\phi(m)$; and the Möbius function $\mu(m)$. The function $\phi(m)/m$ is a strongly multiplicative arithmetic function, a power function $m^s$ is a totally multiplicative arithmetic function.


Comments

The Dirichlet convolution product

$$ (f*g)(n) = \sum_{d\vert n} f(d) g(n/d) $$

yields a group structure on the multiplicative functions. The unit element is given by the function $e$, where $e(1)=1$ and $e(m) = 0$ for all $m > 1$. Another standard multiplicative function is the constant function $\zeta(n)$ with $\zeta(m) = 1$ for all $m$ and its inverse $\mu$, the Möbius function. Note that $\phi = \mu * N_1$, where $N_1(n) = n$ for all $n$, and that $\tau = \zeta * \zeta$, $\sigma = \zeta * N_1$.

Formally, the Dirichlet series of a multiplicative function $f$ has an Euler product:

$$ \sum_{n=1}^\infty f(n) n^{-s} = \prod_p \left({1 + f(p) p^{-s} + f(p^2) p^{-2s} + \cdots }\right) \ , $$

whose form simplifies considerably if $f$ is strongly or totally multiplicative.

References

[a1] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers" , Clarendon Press (1960) pp. Chapts. XVI-XVII
How to Cite This Entry:
Multiplicative arithmetic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiplicative_arithmetic_function&oldid=12136
This article was adapted from an original article by I.P. Kubilyus (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article