Sync C_47 lower bound in README with the constant page - #144
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The table lists the Melas value (11+sqrt(61))/12 ~ 1.5675208, which the 47a page records as the weaker of its two lower bounds: it comes from c_2 >= c_1 monotonicity. The same table's other row, Aldaz 2000 Proposition 1.4 specialized to n = 2, gives 3/4 - sqrt(2)/4 + sqrt(6)/2 ~ 1.6211915, which is strictly larger and so is the better known lower bound. The page's bound tables are chronological and may list bounds inferior to earlier ones, so the last row is not always the best; here the README had followed the last row.
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The README row for$C_{47}$ (centered Hardy–Littlewood maximal constant in dimension 2) lists
as the best known lower bound. The 47a page records that value as the weaker of its two lower bounds — it is Melas' exact$c_1$ carried up to $c_2$ by the monotonicity $c_{d+1}\ge c_d$ . The other row on the same page, Aldaz 2000 Proposition 1.4 specialized to $n=2$ , gives
which is strictly larger, so it is the better lower bound. This syncs the README to it; the page itself already has both rows and needs no change. The upper bound column (4) already matches the page's best.
Numerically:$3/4-\sqrt2/4+\sqrt6/2 = 1.6211914807\ldots$ against $(11+\sqrt{61})/12 = 1.5675208063\ldots$
How I found it: the bound tables are chronological and the template notes they "can include bounds that are inferior to the previous bounds", so the last row is not always the best. I scripted a comparison of every README row against the min (upper) and max (lower) over its page's tables. That flagged three candidates; the other two were false positives I am leaving alone —$C_{52}$ 's 4.453 is conditional on an extra hypothesis, and $C_{81}$ 's 2.175398 is Klyve's RH-conditional bound, so in both cases the README's weaker unconditional value is the right one to show.
One concern only; no other rows touched.