OK, so what the website is doing is taking the actual sum out to a given limit, and then extrapolating it using prime-number-theorem-like arguments. It would be nice to see the raw, unextrapolated sums, which must be monotonically increasing, to confirm that they are all comfortably under the conjectured value. If the raw sums still exceed Lesniak's estimate, then Joerg's objection still stands. It sure looks like Lesniak's conjecture is the result of RIES-like strip-mining. He has about five or six digits of arbitrary constants in his conjecture; if he went through a million settings for those constants, one would roughly expect to be able to match five or six digits of B2. What is giving me pause is that he manages to match 11 digits with only 5 or 6 digits of constants. So if this is a coincidence, it's approaching a one-in-a-million miracle. On Mon, Aug 6, 2018 at 1:12 PM, Joerg Arndt <arndt@jjj.de> wrote:
* Hans Havermann <gladhobo@bell.net> [Aug 06. 2018 18:29]:
AW: "The 'should be around' suggestion can be read as an expectation that the final sum is between ...104 and ...500."
From that website, here are the numerical values calculated by taking the sum to different finite limits:
10^10 1.902160356233... 10^12 1.902160630437... 10^14 1.902160577783... 10^15 1.902160582249... 10^16 1.902160583104...
Note that the value for 10^12 is larger than that for the subsequent terms. That suggests to me that an ever-increasing approximation is not a given. Is that counter-intuitive?
That would indeed nuke my argument.
Best regards, jj
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