0.848969034043...
We can obtain prime(2^d) for d = 0..57 from the b-file for A033844. Given the above result from Robert Price, and letting j_RP = 596765000000, the partial sum through
prime(j_RP) = 17581469834441
is
s(j_RP) = Sum_{k=1..j_RP} 1/(k*prime(k))
= 0.848969034043245206069544346415327714...;
adding to this actual partial sum s(j_RP) the approximate tail value
t(j_RP) =
h'(prime(j_RP), prime(2^40))
+ (Sum_{d=41..57} h'(prime(2^(d-1)), prime(2^d)))
+ lim_{x->infinity} h(prime(2^57), x)
(see the Links entry for an explanation) gives the result 0.84896903404330021273712255895762255... (which seems likely to be correct to at least 20 significant digits).
The table below gives, for j = 2^16, 2^17, ..., 2^32, and j_RP, the actual partial sum s(j) and the sum s(j) + t(j) where t(j) is the approximate tail value beyond prime(j).
.
j s(j) s(j) + t(j)
==== ====================== ======================
2^16 0.84896790758922908159 0.84896903393397518971
2^17 0.84896850050492294891 0.84896903400552099072
2^18 0.84896878057566843770 0.84896903404214147367
2^19 0.84896891330602605081 0.84896903404317536927
2^20 0.84896897639243509768 0.84896903404350431035
2^21 0.84896900645590169648 0.84896903404376063663
2^22 0.84896902081581006534 0.84896903404343742139
2^23 0.84896902768965496764 0.84896903404337393698
2^24 0.84896903098637626311 0.84896903404331189996
2^25 0.84896903257029535468 0.84896903404329806633
2^26 0.84896903333252861584 0.84896903404330030271
2^27 0.84896903369988697984 0.84896903404330084536
2^28 0.84896903387717904236 0.84896903404330042023
2^29 0.84896903396285181513 0.84896903404330024036
2^30 0.84896903400430044877 0.84896903404330021861
2^31 0.84896903402437548991 0.84896903404330021472
2^32 0.84896903403410856545 0.84896903404330021655
... ... ...
j_RP 0.84896903404324520607 0.84896903404330021274
(End)
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