***This is a comment at the comments of Gerhard "Still Computing Oh So Slowly" Paseman's answer, giving just indexes n for more record-holders.***
I used the function ` fharm(n,eps) = frac(h(n))*n^(1+eps) ` to find the local minima for a fixed `eps` and consecutive arguments `n`. The `n-1` of my list should agree with the `n` in the comments. *(The harmonic numbers can be computed in Pari/GP using `h(n) = psi(1+n) + Euler` )*
- Seek manually the first `n` having a local minimum for `w=fharm(n,eps)` and document `n,w`.
- Guess the next `n` by `n=ceil(n*exp(1))+10` compute `w` and decrement `n` as long as also `w=fharm(n,eps)` decreases .
This is what I got using Pari/GP (and 800 digits internal precision, don't know exactly what it needs for the psi-function at this arguments `n`) in a couple of seconds:
n | fharm(n,0.1)
--------------------+----------------
1 0.0
11 0.277901138881
31 1.19066007386
83 0.267050685436
227 1.70522868276
616 1.49190552481
1674 1.70759404228
4550 2.19789655729
12367 1.36471771489
33617 1.68810700160
91380 0.873923093872
248397 1.05774925083
675214 3.52030034776
1835421 2.93703145896
4989191 2.22651939403
13562027 1.04059409912
36865412 0.783097603793
100210581 6.15552338765
272400600 3.07296513521
740461601 2.29064626275
2012783315 2.37389093860
5471312310 1.01950811681
14872568831 3.51618209965
40427833596 2.82622840722
109894245429 10.6250101261
298723530401 7.64454034138
812014744422 6.96806150688
2207284924203 3.81951285918
6000022499693 2.46795156044
16309752131262 12.4771965474
44334502845080 1.85826500223
120513673457548 14.1650514186
327590128640500 21.4949794587
890482293866031 0.75550636057
2420581837980561 21.3699821691
6579823624480555 3.02859881920
17885814992891026 3.21976135398
48618685882356024 1.64093320527
132159290357566703 13.7707827691
359246197441016284 14.6487253696
The set $M$ is here the set of all $n$ where `fharm(n,eps) < 1` . The number of such entries for `n=359246197441016284 ~ 10^17 ` is here `6` . The logarithm of the indexes `n`are
ln(n) ln(n) + gamma
-------------------------------
0.E-809 0.577215664902
2.39789527280 2.97511093770
4.41884060780 4.99605627270
11.4227819151 11.9999975800
17.4227843253 17.9999999902
34.4227843351 35.0000000000
This - and the differences - might be interesting for extrapolating a tendency and a guess for the cardinality of $M$ depending on `eps` and the range of `n` being checked.
Anyway, all numerical test which I've done indicate that the cardinality of $M$ is finite and even decreases with increasing $\varepsilon$ and only if $\varepsilon=0$ might be infinite.
<hr>
This is the Pari/GP-program which I used. Note, that the quotient of two consecutive $n$ approaches $\small \exp(1)$ so I need only the following guess:
Using the function
fharm(n,epsilon=0.1) = frac(psi(1+n)+Euler) * n^(1+epsilon)
the following needs only a couple of steps to find the `n` with the next local minimum when called with `n` having the current local minimum:
{findnext_n(n,eps=0.1)=local(n1,w1,n2,w2);
n1=ceil(n*exp(1)+10); w1 = fharm(n1,eps);
n2=n1-1; w2 = fharm(n2,eps);
while( w2<w1, n1=n2;w1=w2;
n2=n1-1;w2=fharm(n2,eps)
);
return(n1);}
So it is applied:
\\ test it --------------------
eps=0.1
{listlen=40;
list=matrix(listlen,2);
list[1,]=[n=1,fharm(n,eps)];
for(k=2,listlen,
n=findnext_n(n,eps);
list[k,]=[n,fharm(n,eps)];
);
printp(list); }
<hr>
***[Update]***
This is new data for the OP's plot of the cardinality of $M$ by the argument $\varepsilon$ where $\small n <= \lceil(\exp(250-\gamma) \rceil$ :
eps | c=#M | r=c*eps | c= card(M) for n<= approx 10^108
------+------------------
0.01 104 1.040
0.02 55 1.100
0.03 36 1.080
0.04 29 1.160
0.05 21 1.050
0.06 16 0.960
0.07 12 0.840
0.08 12 0.960
0.09 10 0.900
0.10 7 0.700
0.11 6 0.660
0.12 4 0.480
0.13 4 0.520
0.14 4 0.560
0.15 4 0.600
0.16 4 0.640
0.17 4 0.680
0.18 4 0.720
0.19 4 0.760
0.20 4 0.800
<hr>
***[update 2]***
I've extended the search-space for $n$ to the range $\small 1 \ldots e^{1000} \approx 10^{434} $ and to check sanity to the range $\small 1 \ldots e^{2000} \approx 10^{868} $. For epsilons $\varepsilon$ from $0.001$ in $200$ steps up to $0.2$ I made the following graph:
[![picture][1]][1]
For the very small epsilon the increase of the search-space gives slightly higher results, which also illustrates, that for "larger" epsilon the cardinality of $\small M(\varepsilon)$ is finite
***[update 3]*** Uniformity of the `f(n) = frac(h_n) * n^1 ` at the *n*, where `frac(h_n)` has a local minimum (on request of @GerhardPaseman):
[![picture 2][2]][2]
It is also convenient to show a rescaling of the `f(n)` so that we can immediately determine the cardinalities $\small |M(\varepsilon)|$ just by counting the number of dots above $\small e(n)\ge \varepsilon$. The derivation of the formula is
$$ \begin{array}{lll} f(n) \cdot n^\varepsilon &\lt & 1\\
\ln(f(n)) + \varepsilon \cdot \ln(n) &\lt& 0 \\
{\ln(f(n)) \over \ln(n)} + \varepsilon & \lt& 0 \\
\varepsilon & \lt& -\ln_n(f(n)) \end{array}$$
and we simply count, how many dots in the picture show $\small e(n) = - \ln_n(f(n)) \gt \varepsilon$
[![picture][3]][3]
<hr>
# Data for cardinalities
epsilon |M(eps,n)| |M(eps,n)|
n<=e^1000 n<=e^2000
-------------------------------
0.0000 1000 2000
0.0010 633 862
0.0020 430 482
0.0030 315 330
0.0040 258 264
0.0050 204 205
0.0060 171 171
0.0070 151 151
0.0080 135 135
0.0090 120 120
0.0100 109 109
0.0110 100 100
0.0120 98 98
0.0130 89 89
0.0140 84 84
0.0150 79 79
0.0160 73 73
0.0170 69 69
0.0180 62 62
0.0190 57 57
0.0200 55 55
0.0210 50 50
0.0220 47 47
0.0230 44 44
0.0240 42 42
0.0250 41 41
0.0260 39 39
0.0270 39 39
0.0280 37 37
0.0290 37 37
0.0300 36 36
0.0310 36 36
0.0320 36 36
0.0330 36 36
0.0340 34 34
0.0350 34 34
0.0360 34 34
0.0370 33 33
0.0380 32 32
0.0390 31 31
0.0400 29 29
0.0410 29 29
0.0420 29 29
0.0430 26 26
0.0440 26 26
0.0450 25 25
0.0460 25 25
0.0470 24 24
0.0480 24 24
0.0490 23 23
0.0500 21 21
0.0510 21 21
0.0520 21 21
0.0530 20 20
0.0540 20 20
0.0550 20 20
0.0560 20 20
0.0570 19 19
0.0580 18 18
0.0590 18 18
0.0600 16 16
0.0610 16 16
0.0620 16 16
0.0630 16 16
0.0640 16 16
0.0650 16 16
0.0660 16 16
0.0670 16 16
0.0680 15 15
0.0690 14 14
0.0700 12 12
0.0710 12 12
0.0720 12 12
0.0730 12 12
0.0740 12 12
0.0750 12 12
0.0760 12 12
0.0770 12 12
0.0780 12 12
0.0790 12 12
0.0800 12 12
0.0810 11 11
0.0820 11 11
0.0830 11 11
0.0840 11 11
0.0850 11 11
0.0860 11 11
0.0870 11 11
0.0880 10 10
0.0890 10 10
0.0900 10 10
0.0910 10 10
0.0920 10 10
0.0930 10 10
0.0940 10 10
0.0950 10 10
0.0960 9 9
0.0970 9 9
0.0980 8 8
0.0990 8 8
0.1000 7 7
0.1010 7 7
0.1020 7 7
0.1030 7 7
0.1040 7 7
0.1050 7 7
0.1060 7 7
0.1070 7 7
0.1080 7 7
0.1090 6 6
0.1100 6 6
0.1110 6 6
0.1120 5 5
0.1130 5 5
0.1140 5 5
0.1150 4 4
0.1160 4 4
0.1170 4 4
0.1180 4 4
0.1190 4 4
0.1200 4 4
0.1210 4 4
0.1220 4 4
0.1230 4 4
0.1240 4 4
0.1250 4 4
0.1260 4 4
0.1270 4 4
0.1280 4 4
0.1290 4 4
0.1300 4 4
0.1310 4 4
0.1320 4 4
0.1330 4 4
0.1340 4 4
0.1350 4 4
0.1360 4 4
0.1370 4 4
0.1380 4 4
0.1390 4 4
0.1400 4 4
0.1410 4 4
0.1420 4 4
0.1430 4 4
0.1440 4 4
0.1450 4 4
0.1460 4 4
0.1470 4 4
0.1480 4 4
0.1490 4 4
0.1500 4 4
0.1510 4 4
0.1520 4 4
0.1530 4 4
0.1540 4 4
0.1550 4 4
0.1560 4 4
0.1570 4 4
0.1580 4 4
0.1590 4 4
0.1600 4 4
0.1610 4 4
0.1620 4 4
0.1630 4 4
0.1640 4 4
0.1650 4 4
0.1660 4 4
0.1670 4 4
0.1680 4 4
0.1690 4 4
0.1700 4 4
0.1710 4 4
0.1720 4 4
0.1730 4 4
0.1740 4 4
0.1750 4 4
0.1760 4 4
0.1770 4 4
0.1780 4 4
0.1790 4 4
0.1800 4 4
0.1810 4 4
0.1820 4 4
0.1830 4 4
0.1840 4 4
0.1850 4 4
0.1860 4 4
0.1870 4 4
0.1880 4 4
0.1890 4 4
0.1900 4 4
0.1910 4 4
0.1920 4 4
0.1930 4 4
0.1940 4 4
0.1950 4 4
0.1960 4 4
0.1970 4 4
0.1980 4 4
0.1990 4 4
[1]: https://i.sstatic.net/DAz5g.png
[2]: https://i.sstatic.net/18Az3.png
[3]: https://i.sstatic.net/QFKcS.png