Multiplicative function
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
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. 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.
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 = paqb ..., 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:
...
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