Is there any information known for the Ordinary Generating Function for Mobius? $$ \sum_{n=1}^{\infty} {\mu(n)}x^n $$ I know that

It has radius of convergence 1.

Does not have limit as $x\rightarrow 1$.

My question is

Does it have limit as $x\rightarrow \textrm{exp}(i\theta)\neq 1$?

Is there a similar result for $\textrm{exp}(i\theta)\neq 1$ with $$ \sum_{n\leq x}\mu(n)=o(x)$$ i.e. is there a result of this type? $$ \sum_{n\leq x}\mu(n)\textrm{exp}(in\theta) = o(x)$$

EDIT1: Aside from my questions, there are some known results for the OGF for Mobius.

EDIT2: Found a reference for (III)

Reference: JASON P. BELL, NILS BRUIN, AND MICHAEL COONS "TRANSCENDENCE OF GENERATING FUNCTIONS WHOSE COEFFICIENTS ARE MULTIPLICATIVE" available at arXiv:1003.2221v2

P. Borwein, T. Erd´elyi, and F. Littman, Polynomials with coefficients from a finite set available at http://www.cecm.sfu.ca/personal/pborwein/PAPERS/P195.pdf

(I) This function has the unit circle as natural boundary.

(II) This function is a transcendental function.

(III) It is not bounded on any open sector($\{z:|z|<1, z=re^{i\theta}, \alpha<\theta<\beta\}$)of unit disc.

This result (III) together with Baire Category answers my question 1 with a dense set of $\theta$'s, but as Prof Tao mentioned, there can be a possibility that it is bounded for some values of $\theta$.