disclaim: I'm a student major in physics and not in math, with inadequate knowledge of complex analysis, so this answer may have severe mistakes....

It can be understood via Hartogs theorem, at least partially.

Recall that Hartogs extension thm tells us that a complex function with several variables analytic in the connected O\S, where O is open and S is compact, can be extended to S. Then the failure of extension to some subset of S indicates the bad behavior of the whole singularity set.

The main idea is to find some f(z,x) with $f(n,x_0)=H_n$, then try to define the harmonic series as $f(\infty,x_0)$.

**Example1**: $\Sigma_{n=1}^{\infty} \frac{x^n}{n}$, the function here is some extension of $f(z,x)=\Sigma_{n=1}^{n=z}\frac{x^z}{z}$, which can be special value of Lerch function and yet doesn't matter here, (in the following we may use the $z=\infty$ or $x=\infty$ charts implicitly, so you should apply $z \to 1/z$ and something like that) and O is some neighborhood of $(z=\infty,x=1)$. Yet $f(\infty,x)=-\text{Log}(1-x)$ has a brunch point at $x=1$ and a brunch cut running to $x=\infty$, ie, S is noncompact.

**Example2**: $\Sigma_{n=1}^{\infty} \frac{n^x}{n}$, this time the value at $z=\infty$ is zeta function with an isolated pole, yet $f(z,1-x)=\zeta(x)-\zeta(z,x)$, the Riemann zeta is analytic for $x\neq1$, and Hurwitz zeta is usually defined for $z>0$ and $x\neq 1$. Roughly the picture is that f is singular at $x=0$, which is removable, and at a family of x-planes located at {z=negative integers} acumulated around z=infty, thus S is noncompact.

**Example3**: for the $\Sigma_{n=1}^{\infty} \frac{n^x+n^{-x}}{2n}$ regularization appearing in http://math.stackexchange.com/questions/20005/is-it-possible-to-use-regularization-methods-on-the-harmonic-series, the reason is that the singularities are cancelled exactly in pairs and $z=\infty, x=0$ is removable for Hartogs thm.

**Posible relation with renormalization**: the trick here is to choose proper "conter-terms" cancelling the poles exactly - this is the regularization sheme, just like dimensional regularization, but this leaves constant factors unfixed, then the condition $f(n,x_0)=H_n$ comes to rescue - this is alike the renormalization scheme: we use renormalization conditions to connect the regularized results with true values (experimental values). Yet I think it's differnt from other types of resummation methods since substracting poles will change the value of convergent series.