Problem 1107 (difficulty: 8/10)

For \(\displaystyle s>1\) let \(\displaystyle \zeta(s)=\sum\limits_{n=1}^\infty\frac1{n^s}\), \(\displaystyle (p_1,p_2,p_3,\ldots)=(2,3,5,\ldots)\) be the sequence of primes in increasing order.

(a) Prove that \(\displaystyle \displaystyle \lim_{N\to\infty} \prod_{n=1}^N \dfrac1{1-\frac1{p_n^s}} = \zeta(s) \).

(b) Prove that \(\displaystyle \sum\limits_{n=1}^\infty\frac1{p_n}=\infty\).

(c) What is the order of magnitude of \(\displaystyle \displaystyle \sum\limits_{n=1}^\infty \frac1{p_n^s}\) as \(\displaystyle s\to1+0\)?

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