A divisor function σ_{x}(*n*) is an arithmetic function consisting of the sum of the *x*th powers of the positive divisors of *n*:

Notable special cases include , known as *the* divisor function, which gives the number of positive divisors of *n*; and , the sum-of-divisors function, which adds all divisors of *n*, including *n*.

[The aliquot sum, the sum of all proper divisors of *n*, is thus *s*(*n*)=σ(*n*)-*n*; a “perfect” number is one for which *s*(*n*)=*n*.]

For example, consider *n*=12. The divisors are 1,2,3,4,6,12, and so:

.

One can in fact define this for any real, or even complex, *x*. Note that for real *x*>0, we can see by multiplying σ_{–x}(*n*) by *n*^{x}, that

If we have *n*=*a**b*, gcd(*a*,*b*)=1, with *a* having divisors 1,*a*_{1},*a*_{2},…,*a* and *b* having divisors 1,*b*_{1},*b*_{2},…,*b*, then we see that every product *a*_{i}b_{j} is a divisor of *n*, and that every divisor of *n* is such a product. Thus, we can show that divisor functions σ_{x}(*n*) are multiplicative functions.

For powers of primes, we see *p*^{k} has *k*+1 factors: 1,*p*,*p*^{2},…,*p*^{k-1},*p*^{k}. Thus

(for *x*>0).

Considering, then, the prime factorization , we get

.

[Note that , so that the *x*=0 case is the limit of the *x*≠0 general case].

Recall previouly that the Dirichlet series generating function for a sequence consisting of a multiplicative function can be represented as a product over primes:

.

For a divisor function, we see that

.

Now, we need to use the formula for the product of power series:

, where .

Letting *a*_{k}=1, *b*_{k}=*b*^{k} for some positive *b*, then:

. (Note that both series on the left are geometric.)

Now, we see that since , the sum inside the product for our Dirichlet series is equivalent to the right-hand side of the above with and . Thus,

.

Taking the product over the primes, we thus see that

.

This means that

and

.

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Tags: Dirichlet Series Generating Function, Divisor Functions, Divisors, Math, Monday Math, Multiplicative Function, Riemann Zeta Function

This entry was posted on September 14, 2009 at 12:11 am and is filed under Math/Science. You can follow any responses to this entry through the RSS 2.0 feed.
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