Multiplicative function
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 formula_1 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: 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 "r"("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"(1) = 4 ≠ 1. This shows that the function is not multiplicative. However, "r"("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" "q" ..., then "f"("n") = "f"("p") "f"("q") ... This property of multiplicative functions significantly reduces the need for computation, as in the following examples for "n" = 144 = 2 · 3: Similarly, we have: formula_4 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 formula_5, the "Dirichlet convolution" of "f" and "g", by formula_6 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. The Dirichlet convolution of two multiplicative functions is again multiplicative. A proof of this  fact is given by the following expansion for relatively prime formula_15: formula_16 More examples are shown in the article on Dirichlet series. 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 formula_21 on "A" is called multiplicative if formula_22 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  formula_23 where for formula_24 set formula_25 if formula_26 and formula_27 otherwise. The polynomial zeta function is then formula_28 Similar to the situation in , every Dirichlet series of a multiplicative function "h" has a product representation (Euler product): formula_29 where the product runs over all monic irreducible polynomials "P". For example, the product representation of the zeta function is as for the integers: formula_30 Unlike the classical zeta function, formula_31 is a simple rational function: formula_32 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  formula_33 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 formula_34 still holds.