A neat little sum

I was playing around with some infinite summations earlier today and I ended up discovering this little fact that kind of blew my mind a bit :)

1n1+1n21+1n31+=d(1)n+d(2)n2+d(3)n3+\frac{1}{n-1}+\frac{1}{n^2-1}+\frac{1}{n^3-1}+\cdots=\frac{d(1)}{n}+\frac{d(2)}{n^2}+\frac{d(3)}{n^3}+\cdots

Assuming that the sums converge, and where the function d(k)d(k) returns the number of divisors of kk.

Proof: Note that each term on the left hand side is in the form 1nα1\frac{1}{n^{\alpha}-1}. With a little bit of manipulation we can see that:

1nα1=nαnα(nα1)=nα1nα=nα+n2α+n3α+\begin{aligned} \frac{1}{n^{\alpha}-1} &= \frac{n^{-\alpha}}{n^{-\alpha}\left(n^{\alpha}-1\right)} \\ &= \frac{n^{-\alpha}}{1-n^{-\alpha}} \\ &= n^{-\alpha}+n^{-2\alpha}+n^{-3\alpha}+\cdots \end{aligned}

Returning to the original sum, we may now write it as:

n1+n2+n3+n4+n5+n6++n2+n4+n6++n3+n6++n4++n5++n6+\begin{array}{c}&n^{-1} &+ &n^{-2} &+ &n^{-3} &+ &n^{-4} &+ &n^{-5} &+ &n^{-6} &+ &\cdots \\ &+ &n^{-2} & & &+ &n^{-4} & & &+ &n^{-6} &+ &\cdots \\ & & &+ &n^{-3} & & & & &+ &n^{-6} &+ &\cdots \\ & & & & &+ &n^{-4} & & & & &+ &\cdots \\ & & & & & & &+ &n^{-5} & & &+ &\cdots \\ & & & & & & & & &+ &n^{-6} &+ &\cdots \\ & & & & & & & & & & & \vdots & \ddots \end{array}

With the first line representing the expansion of 1n1\frac{1}{n-1}, the second line representing 1n21\frac{1}{n^2-1}, etc. From this, we can see that the initial statement is true. We begin by summing n1n^{-1} to the power of all multiples of 11, then n1n^{-1} to the power of all multiples of 22, and so on. In other words, the number of divisors of the power of n1n^{-1} tells us how many times it appears as a term in the whole expansion.

Has anyone else seen this result before? I'd like to read up on it or other results that are related to it!

Note by Daniel Hinds
7 months ago

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Have a look at this mild generalization:

https://math.stackexchange.com/questions/273275/generating-function-for-the-divisor-function

Patrick Corn - 6 months, 2 weeks ago

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JARID1C is expressed in multiple human tissues, has a paralogue on the Y chromosome and is highly conserved across evolution. Indeed, JARID1C is one of the few genes on the X chromosome escaping X-inactivation. The missense, frameshift and nonsense mutations of JARID1C are associated with X-linked mental retardation (XLMR).

https://www.creative-biogene.com/genesearch/JARID1C.html

Wendy Wilson - 1 month, 2 weeks ago

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