Roots and analytic continuation of $\zeta(s)=\sum_{n>0} \frac{H_n^{-s}}{n} $?

Question

Let

$$\zeta(s)=\sum_{n>0} \frac{H_n^{-s}}{n} $$

Where $H_n$ are the harmonic numbers.

This is well defined for $\Re (s)>1$.

But what about analytic continuation?

And where is

$$\zeta(s) = 0$$

??

Is $\Re(s) = 1$ a natural boundary ?

And can we continue the function beyond $\Re(s) > 1$ ?

Plots are appreciated too :)

Answer 1

$$H_n = \log n+ \gamma + O(1/n)=(\log n+ \gamma)(1+O(1/n))$$ so $$\sum_{n\ge 1} \frac{H_n^{-s} -(\log n+\gamma)^{-s}}n = \sum_{n\ge 1} \frac{(\log n+ \gamma)^{-s}((1+O(1/n))^{-s}-1)}n$$ $$= \sum_{n\ge 1} \frac{(\log n+ \gamma)^{-s}O( s/n)}n$$ is entire.

Beside calling it $\zeta(s)$, I don't think it makes much sense to ask for the zeros, those are notoriously hard to locate for such (generalized) Dirichlet series without additional knowledge, such as an analytic continuation for the inverse..