# Interpolating $G(1)=\sum_{a=1}^{\infty} \frac{1}{a^{a}}$, $G(2) = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{(ab)^{ab}}$ on $\mathbb{C}$

Given that: $$G(1) =\sum_{a=1}^{\infty} \frac{1}{a^{a}}$$

(this is just the Sophomore's dream series, but the rest are not)

$$G(2) = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty} \frac{1}{(ab)^{ab}}$$

$$G(3) = \sum_{a=1}^{\infty} \sum_{b=1}^{\infty}\sum_{c=1}^{\infty} \frac{1}{(abc)^{abc}}$$

I'd like to interpolate $G(z)$ for $z\in\mathbb{C}$ given the above sequence. I can probably compute (with difficulty) $G(n)$ for many $n$.

• Does the sequence above specify a unique $G(z)$?
• Are there analytic tricks which would make finding such an interpolation easy? (the less I have to compute here, the better) *
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