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It's rather well-known that $$ \sum_{p \leq X} \frac{1}{p} \sim \log \log X,$$ where this is a sum over the positive integer primes. Can we efficiently estimate the sum $$ \sum_{p,q \leq X} \frac{1}{pq}$$ where both $p,q$ range over positive primes up to $X$?

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  • $\begingroup$ "Efficiently" is so undefined. Of course we can estimate it... but more important that asymptotic equivalence is the evaluation of the error. $\endgroup$ – Masacroso Jul 16 '15 at 5:45
  • $\begingroup$ I guess we could do $\displaystyle \left(\sum\limits_{p \le \sqrt{x}} \frac{1}{p}\right)^2 \le \sum\limits_{p,q \leq x} \frac{1}{pq} \le \left(\sum\limits_{p \le x} \frac{1}{p}\right)^2$ and say $\displaystyle \sum\limits_{p,q \leq x} \frac{1}{pq} = (\log\log x)^2 + \mathcal{O}(\log \log x)$ $\endgroup$ – r9m Jul 16 '15 at 5:51
  • $\begingroup$ Indeed, the interesting part is the error estimation $\endgroup$ – davidlowryduda Jul 16 '15 at 13:44
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$$\sum_{p,q\leq X}\frac{1}{pq}=\sum_{q\leq X}\frac{1}{q}\sum_{p\leq X}\frac{1}{p}=\left(\log\left(\log\left(X\right)\right)\right)^{2}+O\left(\log\left(\log\left(X\right)\right)\right) $$ from $$\sum_{p\leq X}\frac{1}{p}=\log\left(\log\left(X\right)\right)+O\left(1\right).$$ You can get a better approximation using $$\sum_{p\leq X}\frac{1}{p}=\log\left(\log\left(X\right)\right)+M+O\left(\log^{-1}\left(X\right)\right) $$ where $M $ is the Meissel–Mertens constant.

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An easy consequence of the prime number theorem is that if the real function F(x) is monotonic and is bounded by a polynomial for sufficiently large x, and S(x) is the sum of F(p) over primes p

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(sorry this is my complete answer) An easy consequence of the prime number theorem is that if the real function F(x) is monotonic and is bounded by a polynomial for large x, and if S(x) is the sum of F(p) over primes p less than x, then S(x) is asymptotic to the integral of F(y)/log y , from y=2 to y=x. It also easily extends to functions of more than one variable. Integrating F(y,z)/(log y . log z) from y and z each from 2 to x gives the square of log(log x), to within a constant, which is your answer.

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