The reciprocal sum of the prime-prefix-free numbers converges [pdf] (jdb19937.github.io)
14 points by jdb1729 5 days ago
jdb1729 5 days ago
The reciprocal sum of the prime-prefix-free numbers (https://oeis.org/A287117) converges to a number less than 5*10^14, conditional on the Riemann Hypothesis.
This Lean-verified proof answers a question I posed 10 years ago: https://math.stackexchange.com/questions/2288648/does-the-su...
An equivalent version: if we start with 1 and then output a stream of random bits, reading the number as a big-endian binary number at each step (so each time a bit arrives, the number is multiplied by 2 and 1 is either added or not), the expected time until the number is an odd prime is finite.
gus_massa 4 days ago
Just for reference, the sum of all primes is infinite https://en.wikipedia.org/wiki/Divergence_of_the_sum_of_the_r... so this result is not obvious.
Anyway, I think it's weird it depends on the Riemann Hypothesis.
Do you have some numerical test for intervals like sum up to 1000, up to 10000, up to 100000, up to 1000000, ... ?
jdb1729 4 days ago
Yes, see the table in Remark 7.3 on page 5, it exceeds 3.5, with growth slowing to a crawl. But the calculations mean little, sum(1/p) grows as divergent log(log(n)), so it also has the appearance of convergence on that basis. Many on math.SE argued for divergence (answers since deleted)! Although the proved upper bound is 5e14, heuristically it should be less than 4. I doubt RH is truly necessary. But even relying on RH, the exact value of the sum is elusive.
gus_massa 3 days ago
laichzeit0 5 minutes ago
Ok? Why is this significant?
nextaccountic an hour ago
that's a result that says more about the Riemann hypothesis than this specific problem right?
jdb1729 32 minutes ago
Well, not really, the contrapositive is that if the series diverges the Riemann Hypothesis would be refuted. But few doubt that the Riemann Hypothesis is true. So using it as an assumption merely makes the convergence slightly iffy. For another example of its use, see the deterministic Miller primality test: https://en.wikipedia.org/wiki/Miller–Rabin_primality_test
yzydserd 2 hours ago
“Author of The Da Vinci Code”
?
sorokod 10 minutes ago
That would be Leonardo di ser Piero da Vinci
jdb1729 an hour ago
A nom de OOM.
0976jzhs an hour ago
out of memory?
dash2 an hour ago
Why does it matter to hn? Is it because Dan Brown wrote it?
0976jzhs an hour ago
Because the proof and Lean formalization have been produced by a clanker.