Difference between revisions of "Multiplicative arithmetic function"
(→Comments: Dirichlet convolution of functions corresponds to multiplication of Dirichlet series; Möbius inversion) |
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| − | An [[ | + | An [[arithmetic function]] of one argument, $f(m)$, satisfying the condition |
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f(mn) = f(m) f(n) \label{mult} | f(mn) = f(m) f(n) \label{mult} | ||
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| + | 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 $f(p^a) = f(p)^a$. | ||
| − | + | 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. | |
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| − | Examples of multiplicative arithmetic functions. The function $\tau(m)$, the number of | ||
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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 [[ | + | Formally, the [[Dirichlet series]] of a multiplicative function $f$ has an [[Euler product]]: |
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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 |
Multiplicative arithmetic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiplicative_arithmetic_function&oldid=33870