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
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.
Some multiplicative functions are defined to make formulas easier to write: Other examples of multiplicative functions include many functions of importance in number theory, such as: 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.
In the On-Line Encyclopedia of Integer Sequences, sequences of values of a multiplicative function have the keyword "mult".
A multiplicative function is completely determined by its values at the powers of prime numbers, a consequence of the fundamental theorem of arithmetic.
This property of multiplicative functions significantly reduces the need for computation, as in the following examples for n = 144 = 24 · 32:
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.
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: The Dirichlet convolution can be defined for general arithmetic functions, and yields a ring structure, the Dirichlet ring.
A proof of this fact is given by the following expansion for relatively prime
More examples are shown in the article on Dirichlet series.
if there exists completely multiplicative functions g1,...,gr, h1,...,hs such that
where the inverses are with respect to the Dirichlet convolution.
Rational arithmetical functions of order
is a rational arithmetical function of order
under the Dirichlet convolution is a rational arithmetical function of order
All rational arithmetical functions are multiplicative.
The concept of a rational arithmetical function originates from R. Vaidyanathaswamy (1931).
In 1906, E. Busche stated the identity and, in 1915, S. Ramanujan gave the inverse form for
S. Chowla gave the inverse form for general
The study of Busche-Ramanujan identities begun from an attempt to better understand the special cases given by Busche and Ramanujan.
Totients satisfy a restricted Busche-Ramanujan identity.
Let A = Fq[X], the polynomial ring over the finite field with q elements.
The polynomial zeta function is then Similar to the situation in N, every Dirichlet series of a multiplicative function h has a product representation (Euler product): where the product runs over all monic irreducible polynomials P. For example, the product representation of the zeta function is as for the integers: Unlike the classical zeta function,
is a simple rational function: 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
Multivariate functions can be constructed using multiplicative model estimators.
is quasimultiplicative if there exists a nonzero constant
is semimultiplicative if there exists a nonzero constant
This concept is due to David Rearick (1966).
It is known that the classes of semimultiplicative and Selberg multiplicative functions coincide.