Let for integers $n\geq 1$ the Möbius function denoted as $\mu(n)$. See the definition of this arithmetic function from this MathWorld's article.
I wodered two questions about series involving the Möbius function.
Question.
A) Can you find real numbers $0<\alpha<\beta<1$ satisfing $$\sum_{n=1}^\infty\frac{\mu(n)}{n^3}\sin (n\pi \alpha)=\sum_{n=1}^\infty\frac{\mu(n)}{n^3}\sin (n\pi \beta)=0\,?\tag{1}$$ At least, can we justify that these numbers $\alpha$ and $\beta$ do exist?
B) Can you provide me hints to get an approximation of $$\sup \left\{ \left| \sum_{n=1}^\infty\frac{\mu(n)}{n}\sin(n\pi x) \right|:x\in(0,1) \right\}\,?\tag{2} $$ Thanks you in advance.