Möbius function and prime numbers

1. Sep 15, 2009

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 $$n,x \in [1, 3 \cdot 10^6].$$

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• Plot_Möbius.jpg
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2. Sep 15, 2009

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.

3. Sep 16, 2009

rrronny

Just one question: how is the trend of $$C(n)$$ in the variation of $$n$$?