- #1

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Let [tex]p_i[/tex] denote the i-th prime number. Prove or disprove that:

[tex]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\};[/tex]

[tex] 2)\quad \displaystyle S(n) \sim C \frac {n}{\log{n}},[/tex]

where [tex]C[/tex] is a negative real constant.

In the graph attached are represended the functions [tex]-S(n)[/tex] (red)

and [tex]f (x) : = 0.454353 * x / \log{x}[/tex] (green), with [tex]n,x \in [1, 3 \cdot 10^6].[/tex]

[tex]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\};[/tex]

[tex] 2)\quad \displaystyle S(n) \sim C \frac {n}{\log{n}},[/tex]

where [tex]C[/tex] is a negative real constant.

In the graph attached are represended the functions [tex]-S(n)[/tex] (red)

and [tex]f (x) : = 0.454353 * x / \log{x}[/tex] (green), with [tex]n,x \in [1, 3 \cdot 10^6].[/tex]