I'm trying to define the divisor function $d(k)$ over a larger domain than the integers but the result I produce appears to converge nowhere, including the integer points i originally expected it to Here is my process:
Consider: $$G(x) = \sum_{k=1}^{\infty} \left[ \frac{1}{1-x^k} - 1 \right] = \sum_{i=1}^{\infty} d(i)x^i $$
Observe therefore that $$ d(w) =\frac{1}{ \Gamma(w+1)}\frac{d^{(w)}G}{dx^{(w)}}[0]$$
Note: $$\frac{1}{1-x^k} = \frac{1}{k}\sum_{j=0}^{k-1} \frac{1}{1-e^{2i\pi(\frac{j}{k})}x}$$
$$ d(w) = \frac{1}{\Gamma(w+1)}\frac{d^{(w)}G}{dx^{(w)}} = \frac{1}{\Gamma(w+1)}\sum_{k=1}^{\infty} \left[ \frac{d^{(w)}}{dx^{(w)}} \left[ \frac{1}{k} \sum_{j=0}^{k-1} \frac{1}{1-e^{2i\pi(\frac{j}{k})}x} - 1 \right]\right][0]$$
If we let $w \in \mathbb{N}$ then this simplifies to
$$ d(w) = \frac{1}{\Gamma(w+1)}\sum_{k=1}^{\infty} \left[ \frac{d^{(w)}}{dx^{(w)}} \left[ \frac{1}{k} \sum_{j=0}^{k-1} \frac{1}{1-e^{2i\pi(\frac{j}{k})}x} \right]\right][0]= $$ $$\frac{1}{\Gamma(w+1)}\sum_{k=1}^{\infty} \left[ \frac{1}{k} \sum_{j=0}^{k-1} \frac{(-1)^w \Gamma(w+1)e^{2i\pi w(\frac{j}{k})}}{(1-e^{2i\pi(\frac{j}{k})}x)^w} \right][0]$$
Cancelling out the gamma terms we have
$$ d(w)= \sum_{k=1}^{\infty} \left[ \frac{1}{k} \sum_{j=0}^{k-1} \frac{(-1)^w e^{2i\pi w (\frac{j}{k})}}{(1-e^{2i\pi(\frac{j}{k})}x)^w} \right][0]$$
Now setting $x=0$ yields
$$ d(w) = \sum_{k=1}^{\infty} \left[ \frac{1}{k} \sum_{j=0}^{k-1} (-1)^w e^{2i\pi w (\frac{j}{k})} \right]$$
Factoring out common terms:
$$ d(w) = \sum_{k=1}^{\infty} \left[ \frac{(-1)^w}{k} \sum_{j=0}^{k-1} e^{2i\pi w (\frac{j}{k})} \right]$$
Applying Geometric Series formula:
$$ d(w) = \sum_{k=1}^{\infty} \left[ \frac{(-1)^w}{k} \frac{1 - e^{2 i \pi w}}{1 - e^{\frac{2 wi\pi}{k}}} \right]$$
Which factors out nicely now as
$$ d(w) = (-1)^w (1 - e^{2 wi \pi}) \sum_{k=1}^{\infty} \left[ \frac{1}{k- ke^{\frac{2 wi\pi}{k}}} \right]$$
But it's unclear that this is well defined as the denominator seems to tend to 0 as k tends to infinity qualitatively. Perhaps evaluating this as some type of limit is necessary