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(1)Let $p_n$ be the nth prime number,and $P=\sum_{n=1}^{\infty} \frac{1}{p_{n}}$;
(2)Let $\varphi(n)$ be the Euler's totient function,and $S=\sum_{n=1}^{\infty} \frac{\varphi(n)}{n^{2}}$ .

Walt.White
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    Possible duplicate of: $$$$ https://math.stackexchange.com/questions/15946/does-the-sum-of-reciprocals-of-primes-converge $$$$ https://math.stackexchange.com/questions/3044654/is-it-possible-to-evaluate-the-summation-x-sum-n-1-infty-frac-phinn2 – Matcha Latte Dec 30 '19 at 09:06
  • Welcome to MSE. Your questions are phrased as isolated problems, without any further information or context. This does not match many users' quality standards, so it may attract downvotes, or be put on hold. To prevent that, please [edit] the question. This will help you recognise and resolve the issues. Concretely: please provide context, and include your work and thoughts on the problem. These changes can help in formulating more appropriate answers. – José Carlos Santos Dec 30 '19 at 09:12
  • Thank you! I have already solved the question! – Walt.White Dec 30 '19 at 11:05
  • $\lim_{s\to 1}\log \zeta(s)=\infty$ implies (1) diverges which implies (2) diverges – reuns Dec 30 '19 at 16:33

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