# Möbius function and prime numbers

Let $p_i$ denote the i-th prime number. Prove or disprove that:
$\displaystyle S(n) := \sum_{i=1}^n \mu(p_i + p_{i+1})<0 \quad \forall n \in \mathbb{N}_0 := \left\{1,2,3,...\right\};$
$\displaystyle S(n) \sim C \frac{n}{\log{n}},$
where $C$ is a negative real constant.