Möbius function and prime numbers

[rrronny]

Let p_i denote the i-th prime number. Prove or disprove that:
1)\quad \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\};
2)\quad \displaystyle S(n) \sim C \frac {n}{\log{n}},
where C is a negative real constant.

In the graph attached are represended the functions -S(n) (red)
and f (x) : = 0.454353 * x / \log{x} (green), with n,x \in [1, 3 \cdot 10^6].

 

[CRGreathouse]

The pattern holds up to 4e9 - 21, where S(189961812) = -4681611, which suggests C ~= -0.46979.

But somehow with the failure of the Mertens conjecture I would expect this one to fail eventually.

 

[rrronny]

Hi CRGreathouse, thanks for your reply.
Just one question: how is the trend of C(n) in the variation of n?