weekend lark

Yesterday, I idly wondered about this product:

\prod_{n=1}^{\infty} \log\left(e+\frac{1}{2^{n}}\right)\approx 1.390436264416635011574732\ldots

The obvious generalization is to consider
H(z) = \prod_{n=1}^{\infty} \log\left(e+z^{n}\right) for |z|<1, which looks, after domain coloring, like this:

I think, that besides all the singularities at rational multiples of \pi, that the thing is aggressively one on most of the unit disk

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