Multiplicative functions

In number theory, a multiplicative function is an arithmetic function $f$ of a positive integer $n$ with the property that $f(1)=1$ and
$f(ab) = f(a)f(b)$ whenever $a$ and $b$ are coprime.

An arithmetic function is said to be completely multiplicative (or totally multiplicative) if $f(1)=1$ and $f(ab) = f(a)f(b)$ holds ''for all'' positive integers $a$ and $b$, even when they are not coprime.

Examples

Some multiplicative functions are defined to make formulas easier to write:

* $1(n)$: the constant function defined by $1(n)=1$

* $\operatorname(n)$: the identity function, defined by $\operatorname(n)=n$

* $\operatorname_k(n)$: the power functions, defined by $\operatorname_k(n)=n^k$ for any complex number $k$. As special cases we have
** $\operatorname_0(n)=1(n)$, and
** $\operatorname_1(n)=\operatorname(n)$.

* $\varepsilon(n)$: the function defined by $\varepsilon(n)=1$ if $n=1$ and $0$ otherwise; this is the unit function, so called because it is the multiplicative identity for Dirichlet convolution. Sometimes written as $u(n)$; not to be confused with $\mu(n)$.

* $\lambda(n)$: the Liouville function, $\lambda(n)=(-1)^$, where $\Omega(n)$ is the total number of primes (counted with multiplicity) dividing $n$

The above functions are all completely multiplicative.

* $1_C(n)$: the indicator function of the set $C\subseteq \Z$. This function is multiplicative precisely when $C$ is closed under multiplication of coprime elements. There are also other sets (not closed under multiplication) that give rise to such functions, such as the set of square-free numbers.

Other examples of multiplicative functions include many functions of importance in number theory, such as:

* $\gcd(n,k)$: the greatest common divisor of $n$ and $k$, as a function of $n$, where $k$ is a fixed integer

* $\varphi(n)$: Euler's totient function, which counts the positive integers coprime to (but not bigger than) $n$

* $\mu(n)$: the Möbius function, the parity ($-1$ for odd, $+1$ for even) of the number of prime factors of square-free numbers; $0$ if $n$ is not square-free

* $\sigma_k(n)$: the divisor function, which is the sum of the $k$-th powers of all the positive divisors of $n$ (where $k$ may be any complex number). As special cases we have
** $\sigma_0(n)=d(n)$, the number of positive divisors of $n$,
** $\sigma_1(n)=\sigma(n)$, the sum of all the positive divisors of $n$.

*$\sigma^*_k(n)$: the sum of the $k$-th powers of all unitary divisors of $n$
::$\sigma_k^*(n) \,=\!\!\sum_ \!\!\! d^k$

* $a(n)$: the number of non-isomorphic abelian groups of order $n$

* $\gamma(n)$, defined by $\gamma(n) = (-1)^$, where the additive function $\omega(n)$ is the number of distinct primes dividing $n$
* $\tau(n)$: the Ramanujan tau function
* All Dirichlet characters are completely multiplicative functions, for example
** $(n/p)$, the Legendre symbol, considered as a function of $n$ where $p$ is a fixed prime number

An example of a non-multiplicative function is the arithmetic function $r_2(n)$, the number of representations of $n$ as a sum of squares of two integers, positive, negative, or zero, where in counting the number of ways, reversal of order is allowed. For example:

and therefore $r_2(1)=4\neq 1$. This shows that the function is not multiplicative. However, $r_2(n)/4$ is multiplicative.

In the On-Line Encyclopedia of Integer Sequences, sequences of values of a multiplicative function have the keyword "mult".

See arithmetic function for some other examples of non-multiplicative functions.

Properties

A multiplicative function is completely determined by its values at the powers of prime numbers, a consequence of the fundamental theorem of arithmetic. Thus, if ''n'' is a product of powers of distinct primes, say ''n'' = ''p''^''a'' ''q''^''b'' ..., then
''f''(''n'') = ''f''(''p''^''a'') ''f''(''q''^''b'') ...

This property of multiplicative functions significantly reduces the need for computation, as in the following examples for ''n'' = 144 = 2⁴ · 3²:
$d(144) = \sigma_0(144) = \sigma_0(2^4) \, \sigma_0(3^2) = (1^0 + 2^0 + 4^0 + 8^0 + 16^0)(1^0 + 3^0 + 9^0) = 5 \cdot 3 = 15$
$\sigma(144) = \sigma_1(144) = \sigma_1(2^4) \, \sigma_1(3^2) = (1^1 + 2^1 + 4^1 + 8^1 + 16^1)(1^1 + 3^1 + 9^1) = 31 \cdot 13 = 403$
$\sigma^*(144) = \sigma^*(2^4) \, \sigma^*(3^2) = (1^1 + 16^1)(1^1 + 9^1) = 17 \cdot 10 = 170$

Similarly, we have:
$\varphi(144) = \varphi(2^4) \, \varphi(3^2) = 8 \cdot 6 = 48$

In general, if ''f''(''n'') is a multiplicative function and ''a'', ''b'' are any two positive integers, then

Every completely multiplicative function is a homomorphism of monoids and is completely determined by its restriction to the prime numbers.

Convolution

If ''f'' and ''g'' are two multiplicative functions, one defines a new multiplicative function $f * g$, the '' Dirichlet convolution'' of ''f'' and ''g'', by
$(f \, * \, g)(n) = \sum_ f(d) \, g \left( \frac \right)$
where the sum extends over all positive divisors ''d'' of ''n''.
With this operation, the set of all multiplicative functions turns into an abelian group; the identity element is ''ε''. Convolution is commutative, associative, and distributive over addition.

Relations among the multiplicative functions discussed above include:

* $\mu * 1 = \varepsilon$ (the Möbius inversion formula)
* $(\mu \operatorname_k) * \operatorname_k = \varepsilon$ (generalized Möbius inversion)
* $\varphi * 1 = \operatorname$
* $d = 1 * 1$
* $\sigma = \operatorname * 1 = \varphi * d$
* $\sigma_k = \operatorname_k * 1$
* $\operatorname = \varphi * 1 = \sigma * \mu$
* $\operatorname_k = \sigma_k * \mu$

The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the Dirichlet ring.

The Dirichlet convolution of two multiplicative functions is again multiplicative. A proof of this fact is given by the following expansion for relatively prime $a,b \in \mathbb^$:
$\begin (f \ast g)(ab) & = \sum_ f(d) g\left(\frac\right) \\ &= \sum_ \sum_ f(d_1d_2) g\left(\frac\right) \\ &= \sum_ f(d_1) g\left(\frac\right) \times \sum_ f(d_2) g\left(\frac\right) \\ &= (f \ast g)(a) \cdot (f \ast g)(b). \end$

Dirichlet series for some multiplicative functions

* $\sum_ \frac = \frac$
* $\sum_ \frac = \frac$
* $\sum_ \frac = \frac$
* $\sum_ \frac = \frac$
More examples are shown in the article on Dirichlet series.

Rational arithmetical functions

An arithmetical function ''f'' is said to be a rational arithmetical function of order $(r, s)$ if there exists completely multiplicative functions ''g''_''1'',...,''g''_''r'',
''h''_''1'',...,''h''_''s'' such that
$f=g_1\ast\cdots\ast g_r\ast h_1^\ast\cdots\ast h_s^,$
where the inverses are with respect to the Dirichlet convolution. Rational arithmetical functions of order $(1, 1)$ are known as totient functions, and rational arithmetical functions of order $(2,0)$ are known as quadratic functions or specially multiplicative functions. Euler's function $\varphi(n)$ is a totient function, and the divisor function $\sigma_k(n)$ is a quadratic function.
Completely multiplicative functions are rational arithmetical functions of order $(1,0)$. Liouville's function $\lambda(n)$ is completely multiplicative. The Möbius function $\mu(n)$ is a rational arithmetical function of order $(0, 1)$.
By convention, the identity element $\varepsilon$ under the Dirichlet convolution is a rational arithmetical function of order $(0, 0)$.

All rational arithmetical functions are multiplicative. A multiplicative function ''f'' is a rational arithmetical function of order $(r, s)$ if and only if its Bell series is of the form
$$$$

for all prime numbers $p$.

The concept of a rational arithmetical function originates from R. Vaidyanathaswamy (1931).

Busche-Ramanujan identities

A multiplicative function $f$ is said to be specially multiplicative
if there is a completely multiplicative function $f_A$ such that
:$f(m) f(n) = \sum_ f(mn/d^2) f_A(d)$
for all positive integers $m$ and $n$, or equivalently
:$f(mn) = \sum_ f(m/d) f(n/d) \mu(d) f_A(d)$
for all positive integers $m$ and $n$, where $\mu$ is the Möbius function.
These are known as Busche-Ramanujan identities.
In 1906, E. Busche stated the identity
:$\sigma_k(m) \sigma_k(n) = \sum_ \sigma_k(mn/d^2) d^k,$
and, in 1915, S. Ramanujan gave the inverse form
:$\sigma_k(mn) = \sum_ \sigma_k(m/d) \sigma_k(n/d) \mu(d) d^k$
for $k=0$. S. Chowla gave the inverse form for general $k$ in 1929, see P. J. McCarthy (1986). The study of Busche-Ramanujan identities begun from an attempt to better understand the special cases given by Busche and Ramanujan.

It is known that quadratic functions $f=g_1\ast g_2$ satisfy the Busche-Ramanujan identities with $f_A=g_1g_2$. Quadratic functions are exactly the same as specially multiplicative functions. Totients satisfy a restricted Busche-Ramanujan identity. For further details, see R. Vaidyanathaswamy (1931).

Multiplicative function over

Let , the polynomial ring over the finite field with ''q'' elements. ''A'' is a principal ideal domain and therefore ''A'' is a unique factorization domain.

A complex-valued function $\lambda$ on ''A'' is called multiplicative if $\lambda(fg)=\lambda(f)\lambda(g)$ whenever ''f'' and ''g'' are relatively prime.

Zeta function and Dirichlet series in

Let ''h'' be a polynomial arithmetic function (i.e. a function on set of monic polynomials over ''A''). Its corresponding Dirichlet series is defined to be
: $D_h(s)=\sum_h(f), f, ^,$
where for $g\in A,$ set $, g, =q^$ if $g\ne 0,$ and $, g, =0$ otherwise.

The polynomial zeta function is then
: $\zeta_A(s)=\sum_, f, ^.$

Similar to the situation in , every Dirichlet series of a multiplicative function ''h'' has a product representation ( Euler product):
: $D_(s)=\prod_P \left(\sum_^h(P^), P, ^\right),$
where the product runs over all monic irreducible polynomials ''P''. For example, the product representation of the zeta function is as for the integers:
: $\zeta_A(s)=\prod_(1-, P, ^)^.$

Unlike the classical zeta function, $\zeta_A(s)$ is a simple rational function:
: $\zeta_A(s)=\sum_f , f, ^ = \sum_n\sum_q^=\sum_n(q^)=(1-q^)^.$

In a similar way, If ''f'' and ''g'' are two polynomial arithmetic functions, one defines ''f'' * ''g'', the ''Dirichlet convolution'' of ''f'' and ''g'', by
: $\begin (f*g)(m) &= \sum_ f(d)g\left(\frac\right) \\ &= \sum_f(a)g(b), \end$
where the sum is over all monic divisors ''d'' of ''m'', or equivalently over all pairs (''a'', ''b'') of monic polynomials whose product is ''m''. The identity $D_h D_g = D_$ still holds.

Multivariate

Multivariate functions can be constructed using multiplicative model estimators. Where a matrix function of is defined as $D_N = N^2 \times N(N + 1) / 2$

a sum can be distributed across the product$y_t = \sum(t/T)^u_t = \sum(t/T)^G_t^\epsilon_t$

For the efficient estimation of , the following two nonparametric regressions can be considered: $\tilde^2_t = \frac = \sigma^2(t/T) + \sigma^2(t/T)(\epsilon^2_t - 1),$

and $y^2_t = \sigma^2(t/T) + \sigma^2(t/T)(g_t\epsilon^2_t - 1).$

Thus it gives an estimate value of $L_t(\tau;u) = \sum_^T K_h(u - t/T)\begin ln\tau + \frac \end$

with a local likelihood function for $y^2_t$ with known $g_t$ and unknown $\sigma^2(t/T)$.

Generalizations

An arithmetical function $f$ is
quasimultiplicative if there exists a nonzero constant $c$ such that
$c\,f(mn)=f(m)f(n)$
for all positive integers $m, n$ with $(m, n)=1$. This concept originates by Lahiri (1972).

An arithmetical function $f$ is semimultiplicative
if there exists a nonzero constant $c$, a positive integer $a$ and
a multiplicative function $f_m$ such that
$f(n)=c f_m(n/a)$
for all positive integers $n$
(under the convention that $f_m(x)=0$ if $x$ is not a positive integer.) This concept is due to David Rearick (1966).

An arithmetical function $f$ is Selberg multiplicative if
for each prime $p$ there exists a function $f_p$ on nonnegative integers with $f_p(0)=1$ for
all but finitely many primes $p$ such that
$f(n)=\prod_ f_p(\nu_p(n))$
for all positive integers $n$, where $\nu_p(n)$ is the exponent of $p$ in the canonical factorization of $n$.
See Selberg (1977).

It is known that the classes of semimultiplicative and Selberg multiplicative functions coincide. They both satisfy the arithmetical identity

f(m)f(n)=f((m, n))f( , n

for all positive integers $m, n$. See Haukkanen (2012).

It is well known and easy to see that multiplicative functions are quasimultiplicative functions with $c=1$ and quasimultiplicative functions are semimultiplicative functions with $a=1$.

See also

* Euler product
* Bell series
* Lambert series

References

* See chapter 2 of
* P. J. McCarthy, Introduction to Arithmetical Functions, Universitext. New York: Springer-Verlag, 1986.
*
*
*
*

*

*
*
*S. Ramanujan, Some formulae in the analytic theory of numbers. Messenger 45 (1915), 81--84.

*E. Busche, Lösung einer Aufgabe über Teileranzahlen. Mitt. Math. Ges. Hamb. 4, 229--237 (1906)

*A. Selberg: Remarks on multiplicative functions. Number theory day (Proc. Conf., Rockefeller Univ., New York, 1976), pp. 232–241, Springer, 1977.

External links

*

References

{{Reflist

Number theory