Last week, I talked about the divisor functions σ_{x}(*n*), defined as the sum of the *x*th powers of the divisors of *n*; including the special cases and . I also showed the Dirichlet series generating function

,

with special cases

and .

Now, let us consider . While σ_{a}(*n*)σ_{b}(*n*) is multiplicative, being the product of multiplicative functions (see here), trying to use the Euler product to determine the generating function would prove rather difficult, so instead some number theory will be used. Credit for this method goes to “NonCommAlg” at the Math Help Forum.

First, note that

.

Now, consider the greatest common divisor of our indicies *n* and *m*; if , then we can factor out from , obtaining

.

Next, define . Factoring out from the denominator of each term, we see: . And, our work above on the product of two Riemann zeta functions tells us that . Thus, .

Next, we note that may also be written as , as for each divisor *d* of *n*, is also a divisor of *n*.

Thus, our Dirichlet series is

.

Now, reordering the summations across the indicies, we can sum with *d*_{1} and *d*_{2} over all positive integers, with *n* now the innermost sum; this sum will be over all values of n for which both *d*_{1}|*n* and *d*_{2}|*n*, which is equivalent to , and, therefore, that for some positive integer *m*.

Thus,

.

Now, using the basic number theoretic rule that , we have that , and so

.

Next, we take , and consider reordering the sum above to group terms with the same value of *d*. All those terms with a given *d* will be of the form . Thus, our sum is reordered as

.

You should recognise our inner sum as a form of the function I defined earlier; specifically, the inner sum is equal to . Thus,

.

Letting *a*=*b*=0, we see that

, and more generally, if *a*=*b*,

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

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