Multiplicative function explained

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 $$f(ab)\; =\; f(a)f(b)$$ whenever a and b are coprime.

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.
• The sum of the k-th powers of the Unitary divisors is denoted by σ*_k(n):

\sigma ^{*(n)} _{k} = \sum _{d} d ^{k.}

• 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:

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:$$d(144)\; =\; \backslash sigma\_0(144)\; =\; \backslash sigma\_0(2^4)\; \backslash ,\; \backslash sigma\_0(3^2)\; =\; (1^0\; +\; 2^0\; +\; 4^0\; +\; 8^0\; +\; 16^0)(1^0\; +\; 3^0\; +\; 9^0)\; =\; 5\; \backslash cdot\; 3\; =\; 15$$$$\backslash sigma(144)\; =\; \backslash sigma\_1(144)\; =\; \backslash sigma\_1(2^4)\; \backslash ,\; \backslash sigma\_1(3^2)\; =\; (1^1\; +\; 2^1\; +\; 4^1\; +\; 8^1\; +\; 16^1)(1^1\; +\; 3^1\; +\; 9^1)\; =\; 31\; \backslash cdot\; 13\; =\; 403$$$$\backslash sigma^*(144)\; =\; \backslash sigma^*(2^4)\; \backslash ,\; \backslash sigma^*(3^2)\; =\; (1^1\; +\; 16^1)(1^1\; +\; 9^1)\; =\; 17\; \backslash cdot\; 10\; =\; 170$$

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

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$$(f\; \backslash ,\; *\; \backslash ,\; g)(n)\; =\; \backslash sum\_\; f(d)\; \backslash ,\; g\; \backslash left(\backslash frac\; \backslash right)$$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{Id} _{k)} * \operatorname{Id} _{k} = \varepsilon (generalized Möbius inversion)\varphi * 1 = \operatorname{Id} d = 1 * 1 \sigma = \operatorname{Id} * 1 = \varphi * d \sigma _{k} = \operatorname{Id} _{k} * 1 \operatorname{Id} = \varphi * 1 = \sigma * \mu \operatorname{Id} _{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 Z ^{+} :

Dirichlet series for some multiplicative functions

\sum _{n\ge} (\mu(n))/(n ^{s}) = (1)/(\zeta(s)) \sum _{n\ge} (\varphi(n))/(n ^{s}) = (\zeta(s -1 ))/(\zeta(s)) \sum _{n\ge} (d(n) ^{2})/(n ^{s}) = (\zeta(s) ^{4})/(\zeta( 2 s)) \sum _{n\ge} (2 ^{\omega(n)})/(n ^{s}) = (\zeta(s) ^{2})/(\zeta( 2 s)) 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 g₁,...,g_r, h₁,...,h_s such that $$f=g\_1\backslash ast\backslash cdots\backslash ast\; g\_r\backslash ast\; h\_1^\backslash ast\backslash cdots\backslash ast\; h\_s^,$$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 λ (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 multiplicativeif there is a completely multiplicative function f _{A} such thatf(m) f(n) = \sum _{d\mid} f(mn/d ^{2)} f _{A(d)} for all positive integers m and n , or equivalently f(mn) = \sum _{d\mid} 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 _{d\mid} \sigma ^{2 )} _{k(mn/d} d ^{k,} and, in 1915, S. Ramanujan gave the inverse form\sigma _{k(mn)} = \sum _{d\mid} \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} . In fact, 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

λ on A is called multiplicative if λ (fg) =λ (f) λ (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

$$D\_h(s)=\backslash sum\_h(f)|f|^,$$where for

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

The polynomial zeta function is then

$$\backslash zeta\_A(s)=\backslash sum\_|f|^.$$

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

$$D\_(s)=\backslash prod\_P\; \backslash left(\backslash sum\_^h(P^)|P|^\backslash right),$$where the product runs over all monic irreducible polynomials P. For example, the product representation of the zeta function is as for the integers:

$$\backslash zeta\_A(s)=\backslash prod\_(1-|P|^)^.$$

Unlike the classical zeta function,

\zeta _{A(s)} is a simple rational function:

$$\backslash zeta\_A(s)=\backslash sum\_f\; |f|^\; =\; \backslash sum\_n\backslash sum\_q^=\backslash sum\_n(q^)=(1-q^)^.$$

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

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 _{h*g} still holds.

Multivariate

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

a sum can be distributed across the product$$y\_t\; =\; \backslash sum(t/T)^u\_t\; =\; \backslash sum(t/T)^G\_t^\backslash epsilon\_t$$

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

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

Thus it gives an estimate value of $$L\_t(\backslash tau;u)\; =\; \backslash sum\_^T\; K\_h(u\; -\; t/T)\backslash begin\; ln\backslash tau\; +\; \backslash frac\; \backslash end$$

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 cf(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 anda multiplicative function f _{m} such thatf(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} forall but finitely many primes p such thatf(n) = \prod _{p} 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([m, 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.
• Efficient estimation of a multivariate multiplicative volatility model . Journal of Econometrics . 2010 . 159 . 1 . 55–73 . 10.1016/j.jeconom.2010.04.007 . 54812323 . Hafner . Christian M. . Linton . Oliver .
• P. Haukkanen . Some characterizations of specially multiplicative functions . Int. J. Math. Math. Sci.. 37 . 2335-2344. 2003.
• P. Haukkanen . Extensions of the class of multiplicative functions . East–West Journal of Mathematics. 14 . 2 . 101–113. 2012.
• DB Lahiri. Hypo-multiplicative number-theoretic functions . Aequationes Mathematicae. 8. 3. 316-317 . 1972 .
• D. Rearick . Semi-multiplicative functions . Duke Math. J.. 33. 49-53 . 1966 .
• L. Tóth . Two generalizations of the Busche-Ramanujan identities . International Journal of Number Theory. 9. 1301-1311 . 2013 .
• R. Vaidyanathaswamy . The theory of multiplicative arithmetic functions . Transactions of the American Mathematical Society . 33 . 2 . 579-662 . 1931 . 10.1090/S0002-9947-1931-1501607-1. free .
• 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.