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I was reading through some old analytic number theory notes earlier and found the interesting fact that even though $\sum\frac{1}{p}$ diverges:

$\sum_{\text{known primes}}\frac{1}{p} < 4$.

However these notes were written pre-$2003$. I was wondering if this is still the case. If not then how much bigger has this sum got?

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    $\begingroup$ Not much. $$\sum_{p \leqslant x} \frac{1}{p} = \log \log x + O(1),$$ and $e^{e^4} \approx 5.148435562634557\cdot 10^{23}$. Apart from a handful of huge primes, the "known" primes are all smaller than that. $\endgroup$ – Daniel Fischer Jun 12 '14 at 14:56
  • $\begingroup$ Consider that the number of particles in the universe is about $10^{80}$, and $\log \log 10^{80} \approx 5.2$. $\endgroup$ – John M Jun 12 '14 at 15:08
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    $\begingroup$ The biggest complete list of "small" primes I've seen is up to $10^{10}$. The contribution of the known larger primes to the sum of reciprocals is negligible. So that puts us around $3.2$. $\endgroup$ – John M Jun 12 '14 at 16:38
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    $\begingroup$ Also T.R. Nicely has gone up to $1.6 \times 10^{15}$ summing reciprocals of twin primes, and he's been at that for more than a decade. So if he'd also been summing reciprocals of all primes, we'd be up to around $3.55$. $\endgroup$ – John M Jun 12 '14 at 16:45
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    $\begingroup$ I once attended a talk where the speaker said that the sum of the reciprocals of the known primes was less than 4 --- and always would be. I think this was first suggested by Matiyasevich. (It's 10 years, almost to the day, since I made this observation on another internet math forum. mathforum.org/kb/message.jspa?messageID=581672) $\endgroup$ – Gerry Myerson Jun 21 '14 at 13:49
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Given the growth rate of $\exp \exp x$, I'd say we should be more careful about the contribution of the $O(1)$ term which is about $0.2615$. That means we only need to know primes up to about $1.8 \times 10^{18}$ in order to reach $4$. Numbers in this range are easily sievable to determine primality — you could essentially compute them as fast as you can record them. So the main bottleneck would be storage: let's assume you require a record of all the primes found, rather than just the reciprocal sum (which would be much, much slimmer).

The total amount of storage needed to hold that many primes is about $2.6 \times 10^{18}$ bits, or roughly 320 petabytes. The number of iPhones sold in the last year is about 160 million, so if each of them was allocated a different range of primes, each would only need to hold 2GB of primes, which I doubt would take more than an hour or two of computation.

So basically, depending on how generous you want to be with the definition of "known prime", Apple could break the $4$-barrier with very little effort in the next iOS system update, if they happened to be so inclined :).

Even if you are less generous and require the list of primes to be held in one location, it is physically possible with current high-capacity tape libraries. Apparently an Oracle StorageTek SL8500 can hold 857 PB in a single large enclosure. Just need to find someone who has a spare one lying around with the tapes to fill it...

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    $\begingroup$ My question ignores whether it is feasible to actually compute the sum. Instead I was asking that given a powerful enough computing system should the sum be bigger than $4$ given todays knowledge? $\endgroup$ – fretty Jul 17 '14 at 13:42
  • $\begingroup$ @fretty Given that the sum is feasible to compute, doesn't this answer your latter question affirmatively? $\endgroup$ – Erick Wong Jul 17 '14 at 15:28
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    $\begingroup$ Ah, this is actually interesting. So how high do you estimate we can go feasibly? $\endgroup$ – fretty Jul 18 '14 at 10:00

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