The reciprocal sum of the prime-prefix-free numbers converges [pdf]

The conversation centers on a Lean-verified proof posted by jdb1729 that the reciprocal sum of the prime-prefix-free numbers converges, conditional on the Riemann Hypothesis, with an upper bound of 5×10^14. jdb1729 frames an equivalent probabilistic statement about the expected time for a random growing binary number to become an odd prime and points to numerical data (Remark 7.3) showing the partial sums exceed about 3.5 and then grow very slowly; he also says heuristics suggest the true value should be below 4 and suspects RH might not be essential. Gus_massa highlights that the divergence of the sum of all primes makes the result nontrivial and asks for numerical tests; a jokey aside by yzydserd quips “By: Dan Brown.”

Commenters debate behavior in other bases: gus_massa asks for experiments comparing binary, ternary and quaternary versions and for plots showing differing asymptotics, noting it’s unintuitive that binary could be bounded while nearby bases are not. jdb1729 explains the heuristic criterion he’s read - convergence requires log(b)<1 so only base 2 (since 2<e) yields convergence - and concedes most of the original thinking was done by others, while reiterating that the exact constant remains elusive and the role of RH is still questioned.