Monday Math 89

A divisor function σx(n) is an arithmetic function consisting of the sum of the xth 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 σx(n) by nx, that 

If we have n=ab, gcd(a,b)=1, with a having divisors 1,a1,a2,…,a and b having divisors 1,b1,b2,…,b, then we see that every product aibj is a divisor of n, and that every divisor of n is such a product. Thus, we can show that divisor functions σx(n) are multiplicative functions.
For powers of primes, we see pk has k+1 factors: 1,p,p2,…,pk-1,pk. 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 ak=1, bk=bk 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
.