In number theory, a multiplicative function is an arithmetic function f(n) of a positive integer n with the property that f(1) = 1 and whenever a and b are coprime, then
\( {\displaystyle f(ab)=f(a)f(b).} \)
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)
Idk(n): the power functions, defined by Idk(n) = nk 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) .
1C(n), the indicator function of the set C ⊂ Z, for certain sets C. The indicator function 1C(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
σk(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 = pa qb ..., then f(n) = f(pa) f(qb) ...
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:
φ {\displaystyle \varphi } \varphi (144)= φ {\displaystyle \varphi } \varphi (24) φ {\displaystyle \varphi } \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 Idk) * Idk = ε (generalized Möbius inversion)
\( \varphi \) * 1 = Id
d = 1 * 1
σ = Id * 1 = \( \varphi \) * d
σk = Idk * 1
Id = \( \varphi \) * 1 = σ * μ
Idk = σk * μ
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 \( {\displaystyle a,b\in \mathbb {Z} ^{+}} \) :
\( {\displaystyle (f\ast g)(ab)=\sum _{d|ab}f(d)g\left({\frac {ab}{d}}\right)=\sum _{d_{1}|a}\sum _{d_{2}|b}f(d_{1}d_{2})g\left({\frac {ab}{d_{1}d_{2}}}\right)=\sum _{d_{1}|a}f(d_{1})g\left({\frac {a}{d_{1}}}\right)\times \sum _{d_{2}|b}f(d_{2})g\left({\frac {b}{d_{2}}}\right)=(f\ast g)(a)\cdot (f\ast g)(b).} \)
Dirichlet series for some multiplicative functions
\( \sum _{{n\geq 1}}{\frac {\mu (n)}{n^{s}}}={\frac {1}{\zeta (s)}} \)
\( \sum _{{n\geq 1}}{\frac {\varphi (n)}{n^{s}}}={\frac {\zeta (s-1)}{\zeta (s)}} \)
\( \sum _{{n\geq 1}}{\frac {d(n)^{2}}{n^{s}}}={\frac {\zeta (s)^{4}}{\zeta (2s)}} \)
\( \sum _{{n\geq 1}}{\frac {2^{{\omega (n)}}}{n^{s}}}={\frac {\zeta (s)^{2}}{\zeta (2s)}} \)
More examples are shown in the article on Dirichlet series.
Multiplicative function over Fq[X]
Let A = Fq[X], 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 Fq[X]
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
\( {\displaystyle D_{h}(s)=\sum _{f{\text{ monic}}}h(f)|f|^{-s},} \)
where for \( {\displaystyle g\in A,} \) set \( {\displaystyle |g|=q^{\deg(g)}} \) if \( {\displaystyle g\neq 0,} \) and |g|=0 otherwise.
The polynomial zeta function is then
\( {\displaystyle \zeta _{A}(s)=\sum _{f{\text{ monic}}}|f|^{-s}.} \)
Similar to the situation in N, every Dirichlet series of a multiplicative function h has a product representation (Euler product):
\( {\displaystyle D_{h}(s)=\prod _{P}\left(\sum _{n\mathop {=} 0}^{\infty }h(P^{n})|P|^{-sn}\right),} \)
where the product runs over all monic irreducible polynomials P. For example, the product representation of the zeta function is as for the integers:
\( {\displaystyle \zeta _{A}(s)=\prod _{P}(1-|P|^{-s})^{-1}.} \)
Unlike the classical zeta function,\( {\displaystyle \zeta _{A}(s)} \) is a simple rational function:
\( {\displaystyle \zeta _{A}(s)=\sum _{f}|f|^{-s}=\sum _{n}\sum _{\deg(f)=n}q^{-sn}=\sum _{n}(q^{n-sn})=(1-q^{1-s})^{-1}.}
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
\( {\displaystyle {\begin{aligned}(f*g)(m)&=\sum _{d\mid m}f(d)g\left({\frac {m}{d}}\right)\\&=\sum _{ab=m}f(a)g(b),\end{aligned}}} \)
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 \( {\displaystyle D_{h}D_{g}=D_{h*g}} \) 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
Undergraduate Texts in Mathematics
Graduate Studies in Mathematics
Hellenica World - Scientific Library
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