rrronny
Sep15-09, 10:53 AM
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].
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].