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Question Number 10662 by FilupS last updated on 22/Feb/17
determine if:  (1/2^s )+(1/3^s )+(1/5^s )+...≥(1/1^s )+(1/4^s )+(1/6^s )+...  or:  Σ_(n∈P_(n≥1) ) ^∞ (1/n^s )≥Σ_(n∉P_(n≥1) ) ^∞ (1/n^s ),          Re(s)>1     note:  Σ_(n∈P_(n≥1) ) ^∞ (1/n^s )+Σ_(n∉P_(n≥1) ) ^∞ (1/n^s )=ζ(s)
$$\mathrm{determine}\:\mathrm{if}: \\ $$$$\frac{\mathrm{1}}{\mathrm{2}^{{s}} }+\frac{\mathrm{1}}{\mathrm{3}^{{s}} }+\frac{\mathrm{1}}{\mathrm{5}^{{s}} }+…\geqslant\frac{\mathrm{1}}{\mathrm{1}^{{s}} }+\frac{\mathrm{1}}{\mathrm{4}^{{s}} }+\frac{\mathrm{1}}{\mathrm{6}^{{s}} }+… \\ $$$$\mathrm{or}: \\ $$$$\underset{\underset{{n}\geqslant\mathrm{1}} {{n}\in\mathbb{P}}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{{s}} }\geqslant\underset{\underset{{n}\geqslant\mathrm{1}} {{n}\notin\mathbb{P}}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{{s}} },\:\:\:\:\:\:\:\:\:\:\mathrm{Re}\left({s}\right)>\mathrm{1} \\ $$$$\: \\ $$$$\mathrm{note}: \\ $$$$\underset{\underset{{n}\geqslant\mathrm{1}} {{n}\in\mathbb{P}}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{{s}} }+\underset{\underset{{n}\geqslant\mathrm{1}} {{n}\notin\mathbb{P}}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{{s}} }=\zeta\left({s}\right) \\ $$

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