In number theory, an arithmetic function *f*(*n*) is called a multiplicative function if it possesses the property that *f*(*ab*)=*f*(*a*)*f*(*b*) for any positive integers *a* and *b* which are coprime (gcd(*a*,*b*)=1). Note that since gcd(*n*,1)=1 for any positive integer *n*, we see that for any multiplicative function, *f*(1)=1.

If the property *f*(*ab*)=*f*(*a*)*f*(*b*) holds for all positive integers *a* and *b*, coprime or not, then the function is said to be completely multiplicative.

One example of a multiplicative function is Euler’s totient function *φ*(*n*) (some others may be found here).

Now, consider the prime factorization of *n*:

. Plugging this into a multiplicative function, we see:

.

Thus, a multiplicative function is determined completely by its values for the powers of the prime numbers. This simplifies the computation of multiplicative functions. For example, consider *φ*(*n*). For a power of a prime *p*^{k}, *k*>0, the only positive integers less than or equal to this number which *aren’t* coprime with it are multiples of *p*: *p*, 2*p*, 3*p*, … , (*p*^{k-1})*p*=*p*^{k}; this is *p*^{k-1} numbers, so the totient is

. This allows us to compute the totient of any number from it’s prime factorization, e.g.:

*φ*(720)=*φ*(2^{4}3^{2}5)

=*φ*(2^{4})*φ*(3^{2})*φ*(5)

=2^{4-1}(2-1)·3^{2-1}(3-1)·5^{1-1}(5-1)

=8·6·4=192

Let *f*(*n*) and *g*(*n*) be multiplicative functions, and let *h*(*n*)= *f*(*n*)*g*(*n*) be their product. Then for any coprime positive integers *a* and *b*,

,

so the product of two multiplicative functions is multiplicative.

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Tags: Arithmetic Function, Math, Monday Math, Multiplicative Function, Number Theory, Prime Numbers, Totient

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