We know that $$\sum_{n=1}^{\infty}\cfrac{1}{p_n}$$ diverges. And we know too that $$\sum_{n=1}^{\infty}\cfrac{1}{n^2}$$ converges (to $\frac{\pi^2}{6}$). That means that $$\sum_{n=1}^{\infty}\cfrac{1}{p_n} > \sum_{n=1}^{\infty}\cfrac{1}{n^2}.$$ My question is: Can we conclude from line 3 that $p_n <n^2$?
2026-04-04 06:11:34.1775283094
Can we conclude $p_n < n^2$ from $\zeta(2)$ and Euler's prime theorem?
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