Multiplicative function

Outside number theory, the term multiplicative function is usually used for completely multiplicative functions. This article discusses number theoretic multiplicative functions.

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

f(ab) = f(a) f(b).

An arithmetic function f(n) 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 (completely multiplicative)
• Id(n): identity function, defined by Id(n) = n (completely multiplicative)
• Id_k(n): the power functions, defined by Id_k(n) = n^k for any complex number k (completely multiplicative). As special cases we have
  • Id₀(n) = 1(n) and
  • Id₁(n) = Id(n).
• ε(n): the function defined by ε(n) = 1 if n = 1 and 0 otherwise, sometimes called multiplication unit for Dirichlet convolution or simply the unit function (completely multiplicative). Sometimes written as u(n), but not to be confused with μ(n) .
• 1_C(n), the indicator function of the set C ⊂ Z, for certain sets C. The indicator function 1_C(n) is multiplicative precisely when the set C has the following property for any coprime numbers a and b: the product ab is in C if and only if the numbers a and b are both themselves in C. This is the case if C is the set of squares, cubes, or k-th powers, or if C is 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 $\varphi$, counting the positive integers coprime to (but not bigger than) n
• μ(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
• σ_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). Special cases we have
  • σ₀(n) = d(n) the number of positive divisors of n,
  • σ₁(n) = σ(n), the sum of all the positive divisors of n.
• a(n): the number of non-isomorphic abelian groups of order n.
• λ(n): the Liouville function, λ(n) = (−1)^Ω(n) where Ω(n) is the total number of primes (counted with multiplicity) dividing n. (completely multiplicative).
• γ(n), defined by γ(n) = (−1)^ω(n), where the additive function ω(n) is the number of distinct primes dividing n.
• τ(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 r₂(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:

1 = 1² + 0² = (-1)² + 0² = 0² + 1² = 0² + (-1)²

and therefore r₂(1) = 4 ≠ 1. This shows that the function is not multiplicative. However, r₂(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) = σ₀(144) = σ₀(2⁴)σ₀(3²) = (1⁰ + 2⁰ + 4⁰ + 8⁰ + 16⁰)(1⁰ + 3⁰ + 9⁰) = 5 · 3 = 15,

σ(144) = σ₁(144) = σ₁(2⁴)σ₁(3²) = (1¹ + 2¹ + 4¹ + 8¹ + 16¹)(1¹ + 3¹ + 9¹) = 31 · 13 = 403,

σ^*(144) = σ^*(2⁴)σ^*(3²) = (1¹ + 16¹)(1¹ + 9¹) = 17 · 10 = 170.

Similarly, we have:

$\varphi$(144)=$\varphi$(2⁴)$\varphi$(3²) = 8 · 6 = 48

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

f(a) · f(b) = f(gcd(a,b)) · f(lcm(a,b)).

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

$(f \, * \, g)(n) = \sum_{d|n} f(d) \, g \left( \frac{n}{d} \right)$

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:

• μ * 1 = ε (the Möbius inversion formula)
• (μ Id_k) * Id_k = ε (generalized Möbius inversion)
• $\varphi$ * 1 = Id
• d = 1 * 1
• σ = Id * 1 = $\varphi$ * d
• σ_k = Id_k * 1
• Id = $\varphi$ * 1 = σ * μ
• Id_k = σ_k * μ

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

Dirichlet series for some multiplicative functions

• $\sum_{n\ge 1} \frac{\mu(n)}{n^s} = \frac{1}{\zeta(s)}$
• $\sum_{n\ge 1} \frac{\varphi(n)}{n^s} = \frac{\zeta(s-1)}{\zeta(s)}$
• $\sum_{n\ge 1} \frac{d(n)^2}{n^s} = \frac{\zeta(s)^4}{\zeta(2s)}$
• $\sum_{n\ge 1} \frac{2^{\omega(n)}}{n^s} = \frac{\zeta(s)^2}{\zeta(2s)}$

More examples are shown in the article on Dirichlet series.

Multiplicative function over F_q[X]

Let A=F_q[X], the polynomial ring over the finite field with q elements. A is principal ideal domain and therefore A is 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 F_q[X]

Let h be a polynomial arithmetic function (i.e. a function on set of monic polynomials over A). Its corresponding Dirichlet series define to be

$D_{h}(s)=\sum_{f\text{ monic}}h(f)|f|^{-s}$,

where for $g\in A$, set $|g|=q^{\deg(g)}$ if $g\ne 0$, and $|g|=0$ otherwise.

The polynomial zeta function is then

$\zeta_{A}(s)=\sum_{f\text{ monic}}|f|^{-s}$.

Similar to the situation in N, every Dirichlet series of a multiplicative function h has a product representation (Euler product):

$D_{h}(s)=\prod_{P}(\sum_{n\mathop =0}^{\infty}h(P^{n})|P|^{-sn})$,

Where the product runs over all monic irreducible polynomials P.

For example, the product representation of the zeta function is as for the integers: $\zeta_{A}(s)=\prod_{P}(1-|P|^{-s})^{-1}$.

Unlike the classical zeta function, $\zeta_{A}(s)$ is a simple rational function:

$\zeta_{A}(s)=\sum_{f}(|f|^{-s})=\sum_{n}\sum_{\text{deg(f)=n}}q^{-sn}=\sum_{n}(q^{n-sn})=(1-q^{1-s})^{-1}$.

In a similar way, If ƒ and g are two polynomial arithmetic functions, one defines ƒ * g, the Dirichlet convolution of ƒ and g, by

$\begin{align} (f*g)(m) &= \sum_{d\,\mid \,m} f(d)g\left(\frac{m}{d}\right) \\ &= \sum_{ab\,=\,m}f(a)g(b) \end{align}$

where the sum extends over all monic divisors d of m, or equivalently over all pairs (a, b) of monic polynomials whose product is m. The identity $D_{h}D_{g}=D_{h*g}$ still holds.

See also

• Euler product
• Bell series
• Lambert series

References

• See chapter 2 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathematics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001

External links

• Planet Math