C_81: mark Klyve's upper bound as conditional on GRH - #147
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The table showed 2.175398 [K2007] above 2.288513 [PT2018] with no comments, so the page reads as if the best known upper bound were 2.175398 while the README carries 2.288513. The two are not comparable: [PT2018, section 1] records Klyve's as "under the assumption of the Generalised Riemann Hypothesis we have B < 2.1754", and its own result 1.840503 < B < 2.288513 as unconditional. Both rows now say which they are, and the discrepancy with the README is explained in the comments section rather than left to be rediscovered.
The bound tables are histories, so the Crandall--Pomerance bound B < 2.347 belongs in it: [PT2018, section 1] says "The first upper bound appears to be given by Crandall and Pomerance [5], who showed that B < 2.347", and the 13% improvement claimed in that paper's abstract is measured against it.
The lower-bound rows carried no comments, so nothing distinguished the partial sums recorded here from the extrapolated B ~ 1.902 that is much more widely quoted and is not a bound. Both Nicely references 404 as of 17 August 2026 (the author died in 2019 and his Lynchburg pages are gone). trnicely.net resolves but is now an unrelated site, so it is not a replacement; noted inline so the next reader does not repeat the search.
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What the page says today
The upper-bound table for$C_{81}$ lists
with both comment cells empty, while the README carries$2.288513$ . A reader
comparing the two concludes the README is stale by one row.
It is not. The two bounds are not comparable: [PT2018, §1] states
and the same paper's abstract gives its own$1.840503 < B < 2.288513$ as
unconditional. So the README is right and the page was missing the one fact
that makes it right.
Changes
Three commits, all on
constants/81a.md; no bound values are changed and theREADME needs no edit.
unconditional, and explain in "Additional comments" why the README shows the
larger number.
was missing — [PT2018, §1]: "The first upper bound appears to be given by
Crandall and Pomerance [5], who showed that
claimed in that paper's abstract is measured against it. Rows stay in
chronological order, per CONTRIBUTING.
bounds at all, and which distinguishes them from the extrapolated
One thing I could not fix
Both Nicely references 404:
He died in 2019 and the Lynchburg pages are gone.
trnicely.netstill resolvesbut is now an unrelated commercial site, so it is not a substitute — I checked
before proposing it. archive.org was returning 502/503 while I was working, so I
did not want to paste a snapshot URL I had not actually opened. The dead state is
noted inline instead, so the next reader does not repeat the search; happy to
swap in a Wayback link if you would rather have one.