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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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