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Difference between revisions of "Multiplicative arithmetic function"

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(→‎Comments: Dirichlet convolution of functions corresponds to multiplication of Dirichlet series; Möbius inversion)
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An [[Arithmetic function|arithmetic function]] of one argument, $f(m)$, satisfying the condition
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An [[arithmetic function]] of one argument, $f(m)$, satisfying the condition
  
 
$$
 
$$
 
f(mn) = f(m) f(n) \label{mult}
 
f(mn) = f(m) f(n) \label{mult}
 
$$
 
$$
 +
for any pair of [[coprime numbers|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 \eqref{mult} holds for any two numbers $m,n$, and not just for coprime numbers, then $f$ is called '''totally multiplicative'''; in this case $f(p^a) = f(p)^a$.
  
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 \eqref{mult} holds for any two numbers $m,n$, and not just for coprime numbers, then $f$ is called '''totally multiplicative'''; in this case $f(p^a) = f(p)^a$.
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Examples of multiplicative arithmetic functions. The function $\tau(m)$, the [[number of divisors]] of a natural number $m$; the function $\sigma(m)$, the [[sum of divisors]] of a 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 $m^k$ 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.
 
  
  
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yields a commutative [[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 $\Epsilon(n)$ with $\Epsilon(m) = 1$ 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 $\tau = \Epsilon * \Epsilon$, $\sigma = \Epsilon * N_1$.  In this context, the [[Möbius inversion]] formula states that if $g = \Epsilon * f$ then $f = \mu * g$.
 
yields a commutative [[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 $\Epsilon(n)$ with $\Epsilon(m) = 1$ 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 $\tau = \Epsilon * \Epsilon$, $\sigma = \Epsilon * N_1$.  In this context, the [[Möbius inversion]] formula states that if $g = \Epsilon * f$ then $f = \mu * g$.
  
Formally, the [[Dirichlet series|Dirichlet series]] of a multiplicative function $f$ has an [[Euler product|Euler product]]:
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Formally, the [[Dirichlet series]] of a multiplicative function $f$ has an [[Euler product]]:
  
 
$$
 
$$

Revision as of 17:00, 9 December 2014

2020 Mathematics Subject Classification: Primary: 11A25 [MSN][ZBL]

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

$$ f(mn) = f(m) f(n) \label{mult} $$ 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 \eqref{mult} 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 divisors of a natural number $m$; the function $\sigma(m)$, the sum of divisors of a 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 $m^k$ 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 commutative 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 $\Epsilon(n)$ with $\Epsilon(m) = 1$ 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 $\tau = \Epsilon * \Epsilon$, $\sigma = \Epsilon * N_1$. In this context, the Möbius inversion formula states that if $g = \Epsilon * f$ then $f = \mu * g$.

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. Dirichlet convolution of functions corresponds to multiplication of the associated Dirichlet series.

References

[HaWr] G.H. Hardy, E.M. Wright, "An introduction to the theory of numbers", Clarendon Press (1960) pp. Chapts. XVI-XVII MR2445243 MR1561815 Zbl 0086.25803
How to Cite This Entry:
Multiplicative arithmetic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiplicative_arithmetic_function&oldid=33870
This article was adapted from an original article by I.P. Kubilyus (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article