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I have a rather simple question of number theory which I can't seem to be able to find a good reference for. I am not a specialist and I don't really know where to look. I would like to show that the following sequence

$$ \frac{1}{N^2}\sum_{p \leq N \ p \ \mathrm{prime }}{p^2} $$ goes to infinity. Numerical computation indicates that it is true.

Thank you very much for your attention :)

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    $\begingroup$ I'm voting to close this question as off-topic because it is a duplicate of math.stackexchange.com/questions/49383/… (and the vote to close as duplicate option does not work for posts existing on MSE) $\endgroup$
    – Eric Naslund
    Commented Sep 3, 2015 at 16:47

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This question appeared on Math.stackexhange. The answer linked above uses partial summation to show that for $k>-1$ we have $$\sum_{p\leq x}p^{k}=\text{li}\left(x^{k+1}\right)+O\left(x^{k+1}e^{-c\sqrt{\log x}}\right).$$

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By the prime number theorem, the number of primes between $N/2$ and $N$ is on the order of $N/\log N.$ So, the sum of just those primes squared is of order $N^3/ \log N.$

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    $\begingroup$ "Is on the order of $\log N$" I believe you mean $N/\log N$, and the resulting sum of squares has order $N^3/\log N$. $\endgroup$
    – Eric Naslund
    Commented Sep 3, 2015 at 16:53
  • $\begingroup$ @EricNaslund That's what happens when you don't sleep :( $\endgroup$
    – Igor Rivin
    Commented Sep 3, 2015 at 17:00

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