determine-if-1-2-s-1-3-s-1-5-s-1-1-s-1-4-s-1-6-s-or-n-P-n-1-1-n-s-n-P-n-1-1-n-s-Re-s-gt-1-note-n-P-n-1-1-n-s-n-P-n-

Question Number 10662 by FilupS last updated on 22/Feb/17

determine if :

\frac{1}{2^{s}} + \frac{1}{3^{s}} + \frac{1}{5^{s}} + \dots ⩾ \frac{1}{1^{s}} + \frac{1}{4^{s}} + \frac{1}{6^{s}} + \dots

or :

\underset{\underset{n ⩾ 1}{n \in P}}{\sum^{\infty}} \frac{1}{n^{s}} ⩾ \underset{\underset{n ⩾ 1}{n \notin P}}{\sum^{\infty}} \frac{1}{n^{s}}, Re (s) > 1

note :

\underset{\underset{n ⩾ 1}{n \in P}}{\sum^{\infty}} \frac{1}{n^{s}} + \underset{\underset{n ⩾ 1}{n \notin P}}{\sum^{\infty}} \frac{1}{n^{s}} = ζ (s)