Revisions to Conjecture about harmonic numbers

text again brushed and bugs corrected

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edited Jan 28, 2017 at 22:07

5.3k
1
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38

The set $\small M(0.1,2000)$$\small M(0.1,e^{40})$ (represented by 40 rows) is here the set of all $n$ where $\small f(n,0.1)=w_n \cdot n^{0.1} \lt 1$ . The number of such entries for $\small n=132159290357566703 \approx 10^{17}$ is here 7 .

 {find_n(k)=local(n1,n2,a1,a2);
   n1=floor(exp(k-Euler)); n2=n1+1; a1 = A(n1);a2 = A(n2);
   if(a1<a2,return(n1),return(n2)); 
    }               
         


 \\ create that list one time, then evaluate cardinality for various eps
 \\ using that same list
 {makeList(listlen=40)= local(list,n,w1,logn);
   list=matrix(listlen,4);
   list[1,]=[Euler,0,0,0];
   for(k=2,listlen,
       n=find_n(k); logn=log(n);
       list[k,]=[logn+Euler,n,w1=w(n),-log(w1)/logn];
      );
    return(list); }
    
   \\ compute cardinality with some eps
  cardM(eps,list) = sum(k=1,#list[,1],list[k,4]>eps)     

   \\ apply, note: for long lists we need high internal precision
   list = makeList(40)   \\ the max N is here about e^40
   print(cardM(0.1,list) )

$$ \small{ \begin{array}{lll} A(n) &\lt & {1\over n^1\cdot n^\varepsilon}\\ f(n) \cdot n^\varepsilon &\lt & 1\\ \ln(f(n)) + \varepsilon \cdot \ln(n) &\lt & 0 \\ {\ln(f(n)) \over \ln(n)} + \varepsilon &\lt & 0 \\ \varepsilon & \lt& -\ln_n(f(n)) \end{array} }$$$$ \small{ \begin{array}{lll} A(n) &\lt & {1\over n^1\cdot n^\varepsilon} & \text{ from OP}\\ f(n) \cdot n^\varepsilon &\lt & 1\\ \ln(f(n)) + \varepsilon \cdot \ln(n) &\lt & 0 \\ {\ln(f(n)) \over \ln(n)} + \varepsilon &\lt & 0 \\ \varepsilon & \lt& -\ln_n(f(n)) \end{array} }$$

epsilon  |M(eps,nN)| |M||M(eps,nN)|       
         n<=e^1000N~e^1000  n<=e^2000N~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

The set $\small M(0.1,2000)$ is here the set of all $n$ where $\small f(n,0.1)=w_n \cdot n^{0.1} \lt 1$ . The number of such entries for $\small n=132159290357566703 \approx 10^{17}$ is here 7 .

 {find_n(k)=local(n1,n2,a1,a2);
   n1=floor(exp(k-Euler)); n2=n1+1; a1 = A(n1);a2 = A(n2);
   if(a1<a2,return(n1),return(n2)); 
    }               
         


 \\ create that list one time, then evaluate cardinality for various eps
 \\ using that same list
 {makeList(listlen=40)= local(list,n,w1,logn);
   list=matrix(listlen,4);
   list[1,]=[Euler,0,0,0];
   for(k=2,listlen,
       n=find_n(k); logn=log(n);
       list[k,]=[logn+Euler,n,w1=w(n),-log(w1)/logn];
      );
    return(list); }
    
   \\ compute cardinality with some eps
  cardM(eps,list) = sum(k=1,#list[,1],list[k,4]>eps)     

   \\ apply, note: for long lists we need high internal precision
   list = makeList(40)
   print(cardM(0.1,list) )

$$ \small{ \begin{array}{lll} A(n) &\lt & {1\over n^1\cdot n^\varepsilon}\\ f(n) \cdot n^\varepsilon &\lt & 1\\ \ln(f(n)) + \varepsilon \cdot \ln(n) &\lt & 0 \\ {\ln(f(n)) \over \ln(n)} + \varepsilon &\lt & 0 \\ \varepsilon & \lt& -\ln_n(f(n)) \end{array} }$$

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

The set $\small M(0.1,e^{40})$ (represented by 40 rows) is here the set of all $n$ where $\small f(n,0.1)=w_n \cdot n^{0.1} \lt 1$ . The number of such entries for $\small n=132159290357566703 \approx 10^{17}$ is here 7 .

 {find_n(k)=local(n1,n2,a1,a2);
   n1=floor(exp(k-Euler)); n2=n1+1; a1 = A(n1);a2 = A(n2);
   if(a1<a2,return(n1),return(n2)); 
    }               
         


 \\ create that list one time, then evaluate cardinality for various eps
 \\ using that same list
 {makeList(listlen=40)= local(list,n,w1,logn);
   list=matrix(listlen,4);
   list[1,]=[Euler,0,0,0];
   for(k=2,listlen,
       n=find_n(k); logn=log(n);
       list[k,]=[logn+Euler,n,w1=w(n),-log(w1)/logn];
      );
    return(list); }
    
   \\ compute cardinality with some eps
  cardM(eps,list) = sum(k=1,#list[,1],list[k,4]>eps)     

   \\ apply, note: for long lists we need high internal precision
   list = makeList(40)   \\ the max N is here about e^40
   print(cardM(0.1,list) )

$$ \small{ \begin{array}{lll} A(n) &\lt & {1\over n^1\cdot n^\varepsilon} & \text{ from OP}\\ f(n) \cdot n^\varepsilon &\lt & 1\\ \ln(f(n)) + \varepsilon \cdot \ln(n) &\lt & 0 \\ {\ln(f(n)) \over \ln(n)} + \varepsilon &\lt & 0 \\ \varepsilon & \lt& -\ln_n(f(n)) \end{array} }$$

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

text again brushed and bugs corrected

Source Link

edited Jan 28, 2017 at 21:57

Gottfried Helms

5.3k
1
22
38

Let $h_n$$\small h_n$ denote the n'thn'th harmonic number, $\small A_n=\{h_n\}$ its fractional part. The sequence of $A_n$ has a remarkable shape like a sawtooth-curve with sharp local minimaincreasing wavelength when n is increased and with sharp local minima - a shape which can be exploited when we seek for possible n to be included in M.
I use $\small w_n = \{h_n\} \cdot n$$\small f(n) = w_n = \{h_n\} \cdot n$ and $\small f(n,\varepsilon)=w_n \cdot n^\varepsilon $ with the OP's condition rewritten as $\small f(n,\varepsilon)<1 $ as criterion for the inclusion of n into the set M . Of course the cardinality of M is limited by an upper bound $ \small n \le N$ with some N that can numerically be handled, so actually we should write explicitely $\small |M(\varepsilon,N)| $ instead of M only. I could manage to use $\small N \approx e^{2000}$ withinwith the help of Pari/GP.

The local minima of $\small A_n$ occur near $\small x_k=e^{k-\gamma}, k \in \mathbb N^+$ and the n to be tested is one of the next integers enclosing the $\small x_k$, - so this exact n must be empirically be determined. (Note that my n are the n+1 used in the comments at the OP denoting the previous high value of A )

The harmonic numbers $\small h_n$ can be computed in Pari/GP using h(n) = psi(1+n) + Euler ; however, this seems to be limited to something like $ \small n \lt e^{600}$ and so I had to introduce the Euler-McLaurin-formula for the larger $\small n \gt 1e50$n and implemented the switch from one method to the other at n = 1e50

40*40*

2000*2000*

containing$\small n \approx 10^{840}$ which was possible to compute

just

The basic option according to the focus in the OP's question is now to test $\small w_n \cdot n^\varepsilon \lt 1$, for$\small f(n,\varepsilon) = w_n \cdot n^\varepsilon \lt 1$. For an example with $\small \varepsilon=0.1$ see the 6'th column and the 7'th column allowing to sum to the cardinality of M where we find $\small | M(0.1,N) | = 7$$\small | M(0.1,2000) | = 7$ with $\small n \le N \approx e^{2000}$
A

A second, much nicer, option is to define the function $\small e(n) = -\log_n(w_n) $ and test $\small e(n)>\varepsilon$$\small e(n) \gt \varepsilon$ whether to include this n or not. This is simply possible when looking into the 5'th5'th column and simply compare. This
This latter method allows to compute the cardinalities of $\small M(\varepsilon)$ for arbitrary $\varepsilon$ really fast, for instance to create informative scatter- or lineplots, when first a list for $e(n)$ of length $\small N$ is created and then the successful comparisions with the intended $\varepsilon$ are summed to determine the cardinality.

                n | h_n    | A=frac(h_n) |   w=A*n  |    e(n)   | w*n^0.1|in M?
 -----------------+--------+-------------+----------+-----------+--------+---
                 1  1.00000   10.00000       10.00000     1.000000   10.00000  1
                 4  2.08333   0.0833333     0.333333    0.792481  0.382899  1
                11  3.01988   0.0198773     0.218651    0.634006  0.277901  1
                31  4.02725   0.0272452     0.844601   0.0491822   1.19066  .
                83  5.00207   0.00206827    0.171667    0.398793  0.267051  1
               227  6.00437   0.00436671    0.991243  0.00162136   1.70523  .
               616  7.00127   0.00127410    0.784844   0.0377178   1.49191  .
              1674  8.00049   0.000485572   0.812848   0.0279149   1.70759  .
              4550  9.00021   0.000208063   0.946686  0.00650460   2.19790  .
             12367  10.0000   0.0000430083  0.531883   0.0670005   1.36472  .
             33617  11.0000   0.0000177086  0.595311   0.0497632   1.68811  .
             91380  12.0000   0.00000305167 0.278861    0.111798  0.873923  1
            248397  13.0000   0.00000122948 0.305399   0.0954806   1.05775  .
            675214  14.0000   1.36205E-7    0.919678  0.00623806   3.52030  .
           1835421  15.0000   3.78268E-7    0.694281   0.0252988   2.93703  .
           4989191  16.0000   9.54538E-7    0.476237   0.0481002   2.22652  .
          13562027  17.0000   1.48499E-8    0.201395   0.0975770   1.04059  .
          36865412  18.0000   3.71993E-9    0.137137    0.114033  0.783098  1
         100210581  19.0000   9.73330E-9    0.975380  0.00135314   6.15552  .
         272400600  20.0000   1.61744E-9    0.440592   0.0421997   3.07297  .
         740461601  21.0000   4.01333E-9    0.297172   0.0594162   2.29065  .
        2012783315  22.0000   1.38447E-10   0.278664   0.0596444   2.37389  .
        5471312310  23.0000   1.97920E-11   0.108288   0.0991384   1.01951  .
       14872568831  24.0000   2.27220E-11   0.337935   0.0463183   3.51618  .
       40427833596  25.0000   6.07937E-12   0.245776   0.0574601   2.82623  .
      109894245429  26.0000   7.60776E-12   0.836049  0.00704359   10.6250  .
      298723530401  27.0000   1.82203E-12   0.544283   0.0230213   7.64454  .
      812014744422  28.0000   5.52830E-13   0.448906   0.0292071   6.96806  .
     2207284924203  29.0000   1.00870E-13   0.222650   0.0528504   3.81951  .
     6000022499693  30.0000   2.16954E-14   0.130173   0.0692963   2.46795  .
    16309752131262  31.0000   3.65111E-14   0.595487   0.0170391   12.4772  .
    44334502845080  32.0000   1.81005E-15   0.080247   0.0802804   1.85827  .
   120513673457548  33.0000   4.59281E-15   0.553496   0.0182434   14.1651  .
   327590128640500  34.0000   2.31992E-15   0.759983  0.00821173   21.4950  .
   890482293866031  35.0000   2.71425E-17   0.024169    0.108145  0.755506  1
  2420581837980561  36.0000   2.55560E-16   0.618603   0.0135588   21.3700  .
  6579823624480555  37.0000   1.20561E-17   0.079326   0.0695767   3.02860  .
 17885814992891026  38.0000   4.26642E-18   0.076308   0.0687541   3.21976  .
 48618685882356024  39.0000   7.23781E-19   0.035189   0.0871101   1.64093  .
132159290357566703  40.0000   2.02186E-18   0.267208   0.0334763   13.7708  .

The set $M(0.1)$$\small M(0.1,2000)$ is here the set of all $n$ where $\small w_n \cdot n^{0.1} \lt 1$$\small f(n,0.1)=w_n \cdot n^{0.1} \lt 1$ . The number of such entries for $\small n=132159290357566703 \approx 10^{17}$ is here 7 .
It might be interesting to look at logarithms of the included n . They are

    ln(n)         ln(n) + gamma
  -------------------------------
  0.0       0.57721
  1.38629   1.96351
  2.39789   2.97511
  4.41884   4.99605
  11.4227   12.0000
  17.4227   18.0000
  34.4227   35.0000

This - and the differences - might be interesting for extrapolating a tendency and a guess for the cardinality of $\small M(\varepsilon)$ and the range of n being checked.
Anyway, allAll numerical testtests which I've done indicate that the cardinality of $M$$\small M(\varepsilon \gt 0,\infty)$ is finite and even decreases with increasingroughly reciprocal to $\varepsilon$ and$\small \varepsilon$ and only if $\varepsilon=0$ might beis surely infinite.

Note, that the quotient of two consecutive$n$ approaches$\small \exp(1)$ so I need only the following guess:

  h(n) = Euler+if(n<1e50, psi(1+n),log(n)+1/2/n-1/12/n^2+1/120/n^4-1/252/n^6+1/240/n^8)
  A(n) = if(n==1,return(0)); frac(h(n))
  w(n) = A(n)*n 
  e(n) = if(n==1,return(1)); -log(w(n))/log(n)

 {find_n(k)=local(n1,n2,w1a1,w2a2);
   n1=floor(exp(k-Euler)); n2=n1+1; w1a1 = A(n1);w2;a2 = A(n2);
   if(w1<w2a1<a2,return(n1),return(n2)); 
    }               
         


 \\ create that list --------------------one time, then evaluate cardinality for various eps
 \\ using that same list
 {makeList(listlen=40)= local(list,n,w1,logn);
   list=matrix(listlen,34);
   list[1,]=[Euler,10,0,1];0];
   for(k=2,listlen,
       n=find_n(k); logn=log(n);
       list[k,]=[logn+Euler,n,w1=w(n),-log(w1)/logn];
      );
    return(list); }
    
   \\ compute cardinality with some eps
  cardM(eps,list) = sum(k=1,#list[,1],list[k,4]>eps)     

   \\ apply, note: for long lists we need high internal precision
   list = makeList(40)
   print(cardM(0.1,list) )

***[Update]***

[table 2]: This are sample data for the OP's plot of the cardinality of $\small M(\varepsilon)$ by the argument $\varepsilon$ where $\small N \approx \exp(250) \approx 10^{108}$ :

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<= 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

***[update 2]***I've# Pictures***[Picture 1]***: I've

(Sorry, I had changed the notation of $\small w(n)$ to $\small f(n)$ in that images and formulae) For Remarks: 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

***[Picture 2]***: Indicates uniformity of the$\small f(n) =w_n = frac(h_n) \cdot n^1$ at the *n*, where$\small w_n $ has a local minimum(on request of @GerhardPaseman):[![picture 2][2]][2]

[update[Picture 3] Uniformity of the $\small f(n) = frac(h_n) \cdot n^1$ at the n, where $\small A(n) $ has a local minimum (on request of @GerhardPaseman):

ItIt is also convenient to show a rescaling of the $\small f(n)$ so that we can immediately determine the cardinalities $\small |M(\varepsilon)|$ just by counting the number of dots $\small e(n)$ above $\small e(n)\ge \varepsilon$$\small \varepsilon$. The derivation of the formula is $$ \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

$$ \small{ \begin{array}{lll} A(n) &\lt & {1\over n^1\cdot n^\varepsilon}\\ f(n) \cdot n^\varepsilon &\lt & 1\\ \ln(f(n)) + \varepsilon \cdot \ln(n) &\lt & 0 \\ {\ln(f(n)) \over \ln(n)} + \varepsilon &\lt & 0 \\ \varepsilon & \lt& -\ln_n(f(n)) \end{array} }$$

and we simply count, how many dots in the picture show $\small e(n) = - \ln_n(f(n)) \gt \varepsilon$

-plot

Let $h_n$ denote the n'th harmonic number, $\small A_n=\{h_n\}$ its fractional part. The sequence of $A_n$ has a remarkable shape like a sawtooth-curve with sharp local minima when n is increased which can be exploited when we seek for possible n to be included in M.
I use $\small w_n = \{h_n\} \cdot n$ and $\small f(n,\varepsilon)=w_n \cdot n^\varepsilon $ with the condition $\small f(n,\varepsilon)<1 $ as criterion for the inclusion of n into the set M . Of course the cardinality of M is limited by an upper bound $ \small n \le N$ with some N that can numerically be handled. I could manage to use $\small N \approx e^{2000}$ within Pari/GP.

The local minima of $\small A_n$ occur near $\small x_k=e^{k-\gamma}, k \in \mathbb N^+$ and the n to be tested is one of the next integers enclosing $\small x_k$, so this must be empirically be determined. (Note that my n are the n+1 used in the comments at the OP denoting the previous high value of A )

The harmonic numbers can be computed in Pari/GP using h(n) = psi(1+n) + Euler ; however, this seems to be limited to something like $ \small n \lt e^{600}$ and so I had to introduce the Euler-McLaurin-formula for the larger $\small n \gt 1e50$

40

2000

The basic option is now to test $\small w_n \cdot n^\varepsilon \lt 1$, for an example with $\small \varepsilon=0.1$ see the 6'th column and the 7'th column allowing to sum to the cardinality of M where we find $\small | M(0.1,N) | = 7$ with $\small n \le N \approx e^{2000}$
A second, much nicer, option is to define the function $\small e(n) = -\log_n(w_n) $ and test $\small e(n)>\varepsilon$ whether to include this n or not. This is simply possible looking into the 5'th column. This latter method allows to compute the cardinalities of $\small M(\varepsilon)$ for arbitrary $\varepsilon$ really fast, for instance to create informative scatter- or lineplots.

                n | h_n    | A=frac(h_n) |   w=A*n  |    e(n)   | w*n^0.1|in M?
 -----------------+--------+-------------+----------+-----------+--------+---
                 1  1.00000   1.00000       1.00000     1.000000   1.00000  1
                 4  2.08333   0.0833333     0.333333    0.792481  0.382899  1
                11  3.01988   0.0198773     0.218651    0.634006  0.277901  1
                31  4.02725   0.0272452     0.844601   0.0491822   1.19066  .
                83  5.00207   0.00206827    0.171667    0.398793  0.267051  1
               227  6.00437   0.00436671    0.991243  0.00162136   1.70523  .
               616  7.00127   0.00127410    0.784844   0.0377178   1.49191  .
              1674  8.00049   0.000485572   0.812848   0.0279149   1.70759  .
              4550  9.00021   0.000208063   0.946686  0.00650460   2.19790  .
             12367  10.0000   0.0000430083  0.531883   0.0670005   1.36472  .
             33617  11.0000   0.0000177086  0.595311   0.0497632   1.68811  .
             91380  12.0000   0.00000305167 0.278861    0.111798  0.873923  1
            248397  13.0000   0.00000122948 0.305399   0.0954806   1.05775  .
            675214  14.0000   1.36205E-7    0.919678  0.00623806   3.52030  .
           1835421  15.0000   3.78268E-7    0.694281   0.0252988   2.93703  .
           4989191  16.0000   9.54538E-7    0.476237   0.0481002   2.22652  .
          13562027  17.0000   1.48499E-8    0.201395   0.0975770   1.04059  .
          36865412  18.0000   3.71993E-9    0.137137    0.114033  0.783098  1
         100210581  19.0000   9.73330E-9    0.975380  0.00135314   6.15552  .
         272400600  20.0000   1.61744E-9    0.440592   0.0421997   3.07297  .
         740461601  21.0000   4.01333E-9    0.297172   0.0594162   2.29065  .
        2012783315  22.0000   1.38447E-10   0.278664   0.0596444   2.37389  .
        5471312310  23.0000   1.97920E-11   0.108288   0.0991384   1.01951  .
       14872568831  24.0000   2.27220E-11   0.337935   0.0463183   3.51618  .
       40427833596  25.0000   6.07937E-12   0.245776   0.0574601   2.82623  .
      109894245429  26.0000   7.60776E-12   0.836049  0.00704359   10.6250  .
      298723530401  27.0000   1.82203E-12   0.544283   0.0230213   7.64454  .
      812014744422  28.0000   5.52830E-13   0.448906   0.0292071   6.96806  .
     2207284924203  29.0000   1.00870E-13   0.222650   0.0528504   3.81951  .
     6000022499693  30.0000   2.16954E-14   0.130173   0.0692963   2.46795  .
    16309752131262  31.0000   3.65111E-14   0.595487   0.0170391   12.4772  .
    44334502845080  32.0000   1.81005E-15   0.080247   0.0802804   1.85827  .
   120513673457548  33.0000   4.59281E-15   0.553496   0.0182434   14.1651  .
   327590128640500  34.0000   2.31992E-15   0.759983  0.00821173   21.4950  .
   890482293866031  35.0000   2.71425E-17   0.024169    0.108145  0.755506  1
  2420581837980561  36.0000   2.55560E-16   0.618603   0.0135588   21.3700  .
  6579823624480555  37.0000   1.20561E-17   0.079326   0.0695767   3.02860  .
 17885814992891026  38.0000   4.26642E-18   0.076308   0.0687541   3.21976  .
 48618685882356024  39.0000   7.23781E-19   0.035189   0.0871101   1.64093  .
132159290357566703  40.0000   2.02186E-18   0.267208   0.0334763   13.7708  .

The set $M(0.1)$ is here the set of all $n$ where $\small w_n \cdot n^{0.1} \lt 1$ . The number of such entries for $\small n=132159290357566703 \approx 10^{17}$ is here 7 .
It might be interesting to look at logarithms of the included n . They are

    ln(n)         ln(n) + gamma
  -------------------------------
  0.0       0.57721
  1.38629   1.96351
  2.39789   2.97511
  4.41884   4.99605
  11.4227   12.0000
  17.4227   18.0000
  34.4227   35.0000

This - and the differences - might be interesting for extrapolating a tendency and a guess for the cardinality of $\small M(\varepsilon)$ 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.

Note, that the quotient of two consecutive$n$ approaches$\small \exp(1)$ so I need only the following guess:

  h(n) = Euler+if(n<1e50, psi(1+n),log(n)+1/2/n-1/12/n^2+1/120/n^4-1/252/n^6+1/240/n^8)
  A(n) = frac(h(n))
  w(n) = A(n)*n 
  e(n) = if(n==1,return(1)); -log(w(n))/log(n)

 {find_n(k)=local(n1,n2,w1,w2);
   n1=floor(exp(k-Euler)); n2=n1+1; w1 = A(n1);w2 = A(n2);
   if(w1<w2,return(n1),return(n2)); 
    }               
         


 \\ create list --------------------
 {makeList(listlen=40)= local(list,n,w1,logn);
   list=matrix(listlen,3);
   list[1,]=[Euler,1,0,1];
   for(k=2,listlen,
       n=find_n(k); logn=log(n);
       list[k,]=[logn+Euler,n,w1=w(n),-log(w1)/logn];
      );
    return(list); }
    
   \\ compute cardinality with some eps
  cardM(eps,list) = sum(k=1,#list[,1],list[k,4]>eps)     

   \\ apply, note: for long lists we need high internal precision
   list = makeList(40)
   print(cardM(0.1,list) )

***[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

***[update 2]***I've

(Sorry, I had changed the notation of $\small w(n)$ to $\small f(n)$ in that images and formulae) 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 $\small f(n) = frac(h_n) \cdot n^1$ at the n, where $\small A(n) $ has a local minimum (on request of @GerhardPaseman):

It is also convenient to show a rescaling of the $\small f(n)$ 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 $\small e(n) = - \ln_n(f(n)) \gt \varepsilon$

Let $\small h_n$ denote the n'th harmonic number, $\small A_n=\{h_n\}$ its fractional part. The sequence of $A_n$ has a remarkable shape like a sawtooth-curve with increasing wavelength when n is increased and with sharp local minima - a shape which can be exploited when we seek for possible n to be included in M.
I use $\small f(n) = w_n = \{h_n\} \cdot n$ and $\small f(n,\varepsilon)=w_n \cdot n^\varepsilon $ with the OP's condition rewritten as $\small f(n,\varepsilon)<1 $ as criterion for the inclusion of n into the set M . Of course the cardinality of M is limited by an upper bound $ \small n \le N$ with some N that can numerically be handled, so actually we should write explicitely $\small |M(\varepsilon,N)| $ instead of M only. I could manage to use $\small N \approx e^{2000}$ with the help of Pari/GP.

The local minima of $\small A_n$ occur near $\small x_k=e^{k-\gamma}, k \in \mathbb N^+$ and the n to be tested is one of the next integers enclosing the $\small x_k$ - so this exact n must be empirically be determined. (Note that my n are the n+1 used in the comments at the OP denoting the previous high value of A )

The harmonic numbers $\small h_n$ can be computed in Pari/GP using h(n) = psi(1+n) + Euler ; however, this seems to be limited to something like $ \small n \lt e^{600}$ and so I had to introduce the Euler-McLaurin-formula for the larger n and implemented the switch from one method to the other at n = 1e50

*40*

*2000*

containing$\small n \approx 10^{840}$ which was possible to compute

just

The basic option according to the focus in the OP's question is to test $\small f(n,\varepsilon) = w_n \cdot n^\varepsilon \lt 1$. For an example with $\small \varepsilon=0.1$ see the 6'th column and the 7'th column allowing to sum to the cardinality of M where we find $\small | M(0.1,2000) | = 7$ with $\small n \le N \approx e^{2000}$

A second, much nicer, option is to define the function $\small e(n) = -\log_n(w_n) $ and test $\small e(n) \gt \varepsilon$ whether to include this n or not. This is simply possible when looking into the 5'th column and simply compare.
This latter method allows to compute the cardinalities of $\small M(\varepsilon)$ for arbitrary $\varepsilon$ really fast, for instance to create informative scatter- or lineplots, when first a list for $e(n)$ of length $\small N$ is created and then the successful comparisions with the intended $\varepsilon$ are summed to determine the cardinality.

                n | h_n    | A=frac(h_n) |   w=A*n  |    e(n)   | w*n^0.1|in M?
 -----------------+--------+-------------+----------+-----------+--------+---
                 1  1.00000   0.00000       0.00000     1.000000   0.00000  1
                 4  2.08333   0.0833333     0.333333    0.792481  0.382899  1
                11  3.01988   0.0198773     0.218651    0.634006  0.277901  1
                31  4.02725   0.0272452     0.844601   0.0491822   1.19066  .
                83  5.00207   0.00206827    0.171667    0.398793  0.267051  1
               227  6.00437   0.00436671    0.991243  0.00162136   1.70523  .
               616  7.00127   0.00127410    0.784844   0.0377178   1.49191  .
              1674  8.00049   0.000485572   0.812848   0.0279149   1.70759  .
              4550  9.00021   0.000208063   0.946686  0.00650460   2.19790  .
             12367  10.0000   0.0000430083  0.531883   0.0670005   1.36472  .
             33617  11.0000   0.0000177086  0.595311   0.0497632   1.68811  .
             91380  12.0000   0.00000305167 0.278861    0.111798  0.873923  1
            248397  13.0000   0.00000122948 0.305399   0.0954806   1.05775  .
            675214  14.0000   1.36205E-7    0.919678  0.00623806   3.52030  .
           1835421  15.0000   3.78268E-7    0.694281   0.0252988   2.93703  .
           4989191  16.0000   9.54538E-7    0.476237   0.0481002   2.22652  .
          13562027  17.0000   1.48499E-8    0.201395   0.0975770   1.04059  .
          36865412  18.0000   3.71993E-9    0.137137    0.114033  0.783098  1
         100210581  19.0000   9.73330E-9    0.975380  0.00135314   6.15552  .
         272400600  20.0000   1.61744E-9    0.440592   0.0421997   3.07297  .
         740461601  21.0000   4.01333E-9    0.297172   0.0594162   2.29065  .
        2012783315  22.0000   1.38447E-10   0.278664   0.0596444   2.37389  .
        5471312310  23.0000   1.97920E-11   0.108288   0.0991384   1.01951  .
       14872568831  24.0000   2.27220E-11   0.337935   0.0463183   3.51618  .
       40427833596  25.0000   6.07937E-12   0.245776   0.0574601   2.82623  .
      109894245429  26.0000   7.60776E-12   0.836049  0.00704359   10.6250  .
      298723530401  27.0000   1.82203E-12   0.544283   0.0230213   7.64454  .
      812014744422  28.0000   5.52830E-13   0.448906   0.0292071   6.96806  .
     2207284924203  29.0000   1.00870E-13   0.222650   0.0528504   3.81951  .
     6000022499693  30.0000   2.16954E-14   0.130173   0.0692963   2.46795  .
    16309752131262  31.0000   3.65111E-14   0.595487   0.0170391   12.4772  .
    44334502845080  32.0000   1.81005E-15   0.080247   0.0802804   1.85827  .
   120513673457548  33.0000   4.59281E-15   0.553496   0.0182434   14.1651  .
   327590128640500  34.0000   2.31992E-15   0.759983  0.00821173   21.4950  .
   890482293866031  35.0000   2.71425E-17   0.024169    0.108145  0.755506  1
  2420581837980561  36.0000   2.55560E-16   0.618603   0.0135588   21.3700  .
  6579823624480555  37.0000   1.20561E-17   0.079326   0.0695767   3.02860  .
 17885814992891026  38.0000   4.26642E-18   0.076308   0.0687541   3.21976  .
 48618685882356024  39.0000   7.23781E-19   0.035189   0.0871101   1.64093  .
132159290357566703  40.0000   2.02186E-18   0.267208   0.0334763   13.7708  .

The set $\small M(0.1,2000)$ is here the set of all $n$ where $\small f(n,0.1)=w_n \cdot n^{0.1} \lt 1$ . The number of such entries for $\small n=132159290357566703 \approx 10^{17}$ is here 7 .

All numerical tests which I've done indicate that the cardinality of $\small M(\varepsilon \gt 0,\infty)$ is finite and roughly reciprocal to $\small \varepsilon$ and only if $\varepsilon=0$ is surely infinite.

  h(n) = Euler+if(n<1e50, psi(1+n),log(n)+1/2/n-1/12/n^2+1/120/n^4-1/252/n^6+1/240/n^8)
  A(n) = if(n==1,return(0)); frac(h(n))
  w(n) = A(n)*n 
  e(n) = if(n==1,return(1)); -log(w(n))/log(n)

 {find_n(k)=local(n1,n2,a1,a2);
   n1=floor(exp(k-Euler)); n2=n1+1; a1 = A(n1);a2 = A(n2);
   if(a1<a2,return(n1),return(n2)); 
    }               
         


 \\ create that list one time, then evaluate cardinality for various eps
 \\ using that same list
 {makeList(listlen=40)= local(list,n,w1,logn);
   list=matrix(listlen,4);
   list[1,]=[Euler,0,0,0];
   for(k=2,listlen,
       n=find_n(k); logn=log(n);
       list[k,]=[logn+Euler,n,w1=w(n),-log(w1)/logn];
      );
    return(list); }
    
   \\ compute cardinality with some eps
  cardM(eps,list) = sum(k=1,#list[,1],list[k,4]>eps)     

   \\ apply, note: for long lists we need high internal precision
   list = makeList(40)
   print(cardM(0.1,list) )

[table 2]: This are sample data for the OP's plot of the cardinality of $\small M(\varepsilon)$ by the argument $\varepsilon$ where $\small N \approx \exp(250) \approx 10^{108}$ :

 eps   | c=#M | r=c*eps  | c= card(M) for n<= 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

# Pictures***[Picture 1]***: I've

Remarks: 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

***[Picture 2]***: Indicates uniformity of the$\small f(n) =w_n = frac(h_n) \cdot n^1$ at the *n*, where$\small w_n $ has a local minimum(on request of @GerhardPaseman):[![picture 2][2]][2]

[Picture 3]:It is also convenient to show a rescaling of the $\small f(n)$ so that we can immediately determine the cardinalities $\small |M(\varepsilon)|$ just by counting the number of dots $\small e(n)$ above $\small \varepsilon$. The derivation of the formula is

$$ \small{ \begin{array}{lll} A(n) &\lt & {1\over n^1\cdot n^\varepsilon}\\ f(n) \cdot n^\varepsilon &\lt & 1\\ \ln(f(n)) + \varepsilon \cdot \ln(n) &\lt & 0 \\ {\ln(f(n)) \over \ln(n)} + \varepsilon &\lt & 0 \\ \varepsilon & \lt& -\ln_n(f(n)) \end{array} }$$

and we simply count, how many dots in the picture show $\small e(n) = - \ln_n(f(n)) \gt \varepsilon$

-plot

first part of text massively straightened

Source Link

edited Jan 28, 2017 at 13:34

Gottfried Helms

5.3k
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38

I used the functionLet fharm(n,eps) = frac(h(n))*n^(1+eps) to find$h_n$ denote the n'th harmonic number, $\small A_n=\{h_n\}$ its fractional part. The sequence of $A_n$ has a remarkable shape like a sawtooth-curve with sharp local minima when n is increased which can be exploited when we seek for a fixedpossible eps and consecutive argumentsn to be included in nM. The
I use n-1 of my list should agree$\small w_n = \{h_n\} \cdot n$ and $\small f(n,\varepsilon)=w_n \cdot n^\varepsilon $ with the condition n in$\small f(n,\varepsilon)<1 $ as criterion for the commentsinclusion of n into the set M . Of course the cardinality of (The harmonic numbers can be computed in Pari/GP using h(n) = psi(1+n) + Euler )M is limited by an upper bound $ \small n \le N$ with some N that can numerically be handled. I could manage to use $\small N \approx e^{2000}$ within Pari/GP.

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 .

The local minima of $\small A_n$ occur near $\small x_k=e^{k-\gamma}, k \in \mathbb N^+$ and the n to be tested is one of the next integers enclosing $\small x_k$, so this must be empirically be determined. (Note that my n are the n+1 used in the comments at the OP denoting the previous high value of A )

This is what I got usingThe harmonic numbers can be computed in Pari/GP using (and 800 digits internal precisionh(n) = psi(1+n) + Euler ; however, don't know exactly what it needs forthis seems to be limited to something like $ \small n \lt e^{600}$ and so I had to introduce the psiEuler-function atMcLaurin-formula for the larger $\small n \gt 1e50$

The following table shows the first 40 entries of my 2000-row table using Pari/GP in a couple of seconds.

The basic option is now to test $\small w_n \cdot n^\varepsilon \lt 1$, for an example with $\small \varepsilon=0.1$ see the 6'th column and the 7'th column allowing to sum to the cardinality of M where we find $\small | M(0.1,N) | = 7$ with $\small n \le N \approx e^{2000}$
A second, much nicer, option is to define the function $\small e(n) = -\log_n(w_n) $ and test $\small e(n)>\varepsilon$ whether to include this arguments n) in a couplen or not. This is simply possible looking into the 5'th column. This latter method allows to compute the cardinalities of seconds:$\small M(\varepsilon)$ for arbitrary $\varepsilon$ really fast, for instance to create informative scatter- or lineplots.

                n | h_n    | A=frac(h_n) | fharm  w=A*n  |    e(n,0.1)   | w*n^0.1|in M?
 -----------------+--------+-------------+----------+-----------+--------+---
        
          1  1.00000   1.00000       1.00000     1.000000   1.00000  1
                 4  2.08333   0.0833333     0.333333 
    0.792481  0.382899  1
                11  3.01988   0.277901138881
0198773     0.218651    0.634006  0.277901  1
                31  4.02725   0.0272452     0.844601   0.0491822   1.19066007386
19066  .
                83  5.00207   0.267050685436
00206827    0.171667    0.398793  0.267051  1
               227  6.00437   0.00436671    0.991243  0.00162136   1.70522868276
70523  .
               616  7.00127   0.00127410    0.784844   0.0377178   1.49190552481
49191  .
              1674  8.00049   0.000485572   0.812848   0.0279149   1.70759404228
70759  .
              4550  9.00021   0.000208063   0.946686  0.00650460   2.19789655729
19790  .
             12367  10.0000   0.0000430083  0.531883   0.0670005   1.36471771489
36472  .
             33617  11.0000   0.0000177086  0.595311   0.0497632   1.68810700160
68811  .
             91380  12.0000   0.873923093872
00000305167 0.278861    0.111798  0.873923  1
            248397  13.0000   0.00000122948 0.305399   0.0954806   1.05774925083
05775  .
            675214  14.0000   1.36205E-7    0.919678  0.00623806   3.52030034776
52030  .
           1835421  15.0000   3.78268E-7    0.694281   0.0252988   2.93703145896
93703  .
           4989191  16.0000   9.54538E-7    0.476237   0.0481002   2.22651939403
22652  .
          13562027  17.0000   1.04059409912
48499E-8    0.201395   0.0975770   1.04059  .
          36865412  18.0000   3.71993E-9    0.783097603793
137137    0.114033  0.783098  1
         100210581  19.0000   9.73330E-9    0.975380  0.00135314   6.15552338765
15552  .
         272400600  20.0000   1.61744E-9    0.440592   0.0421997   3.07296513521
07297  .
         740461601  21.0000   4.01333E-9    0.297172   0.0594162   2.29064626275
29065  .
        2012783315  22.0000   1.38447E-10   0.278664   0.0596444   2.37389093860
37389  .
        5471312310  23.0000   1.01950811681
97920E-11   0.108288   0.0991384   1.01951  .
       14872568831  24.0000   2.27220E-11   0.337935   0.0463183   3.51618209965
51618  .
       40427833596  25.0000   6.07937E-12   0.245776   0.0574601   2.82622840722
82623  .
      109894245429  26.0000   7.60776E-12   0.836049  0.00704359   10.6250101261
6250  .
      298723530401  27.0000   1.82203E-12   0.544283   0.0230213   7.64454034138
64454  .
      812014744422  28.0000   5.52830E-13   0.448906   0.0292071   6.96806150688
96806  .
     2207284924203  29.0000   1.00870E-13   0.222650   0.0528504   3.81951285918
81951  .
     6000022499693  30.0000   2.46795156044
16954E-14   0.130173   0.0692963   2.46795  .
    16309752131262  31.0000   3.65111E-14   0.595487   0.0170391   12.4771965474
4772  .
    44334502845080  32.0000   1.85826500223
81005E-15   0.080247   0.0802804   1.85827  .
   120513673457548  33.0000   4.59281E-15   0.553496   0.0182434   14.1650514186
1651  .
   327590128640500  34.0000   2.31992E-15   0.759983  0.00821173   21.4949794587
4950  .
   890482293866031  35.0000   2.71425E-17   0.75550636057
024169    0.108145  0.755506  1
  2420581837980561  36.0000   2.55560E-16   0.618603   0.0135588   21.3699821691
3700  .
  6579823624480555  37.0000   1.20561E-17   0.079326   0.0695767   3.02859881920
02860  .
 17885814992891026  38.0000   4.26642E-18   0.076308   0.0687541   3.21976135398
21976  .
 48618685882356024  39.0000   7.23781E-19   0.035189   0.0871101   1.64093320527
64093  .
132159290357566703  40.0000 13  2.7707827691
02186E-18  359246197441016284 0.267208   140.64872536960334763   13.7708  .

The set $M$$M(0.1)$ is here the set of all $n$ where fharm(n,eps) < 1$\small w_n \cdot n^{0.1} \lt 1$ . The number of such entries for n=359246197441016284 ~ 10^17 $\small n=132159290357566703 \approx 10^{17}$ is here 67 . The logarithm
It might be interesting to look at logarithms of the indexesincluded naren . They are

    ln(n)         ln(n) + gamma
  -------------------------------
  0.0       0.E-80957721
  01.57721566490238629   1.96351
  2.3978952728039789   2.9751109377097511
  4.4188406078041884   4.9960562727099605
  11.42278191514227   1112.99999758000000
  17.42278432534227   1718.99999999020000
  34.42278433514227   35.00000000000000

This - and the differences - might be interesting for extrapolating a tendency and a guess for the cardinality of $M$ depending on eps$\small M(\varepsilon)$ and the range of nn 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.

This# Pari/GP toolsThis

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:functions

{findnext_n  h(n,eps=0.1)=local = Euler+if(n1,w1,n2n<1e50,w2);
 n1=ceilpsi(n*exp1+n),log(1n)+10+1/2/n-1/12/n^2+1/120/n^4-1/252/n^6+1/240/n^8); w1 
 = fharmA(n1,epsn);
 n2=n1-1;              w2 = fharmfrac(n2,epsh(n);)
 while( w2<w1, w(n) = n1=n2;w1=w2;A(n)*n 
               e(n) = n2=n1-1;w2=fharmif(n2n==1,epsreturn(1)
      );
  return-log(n1w(n);}               
         )/log(n)

So it is appliedThe following needs only two steps to find the n with the next local minimum when called with index $\small k \in \mathbb N$ :

 {find_n(k)=local(n1,n2,w1,w2);
   n1=floor(exp(k-Euler)); n2=n1+1; w1 = A(n1);w2 = A(n2);
   if(w1<w2,return(n1),return(n2)); 
    }               
         


 \\ testcreate itlist --------------------
 eps=0.1
 {listlen=40;makeList(listlen=40)= local(list,n,w1,logn);
   list=matrix(listlen,23);
   list[1,]=[n=1]=[Euler,fharm(n1,eps)];0,1];
   for(k=2,listlen,
       n=findnext_nn=find_n(k); logn=log(n,eps);
       list[k,]=[n]=[logn+Euler,fharmn,w1=w(n),eps-log(w1)];/logn];
      );
 printp   return(list); }
    
   \\ compute cardinality with some eps
  cardM(eps,list) = sum(k=1,#list[,1],list[k,4]>eps)     

   \\ apply, note: for long lists we need high internal precision
   list = makeList(40)
   print(cardM(0.1,list) )

For(Sorry, I had changed the notation of $\small w(n)$ to $\small f(n)$ in that images and formulae) 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 $\small f(n) = frac(h_n) \cdot n^1$ at the n, where frac(h_n)$\small A(n) $ has a local minimum (on request of @GerhardPaseman):

It is also convenient to show a rescaling of the f(n)$\small f(n)$ 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 $\small e(n) = - \ln_n(f(n)) \gt \varepsilon$

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 nare

    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.

This

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); }

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):

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 $\small e(n) = - \ln_n(f(n)) \gt \varepsilon$

Let $h_n$ denote the n'th harmonic number, $\small A_n=\{h_n\}$ its fractional part. The sequence of $A_n$ has a remarkable shape like a sawtooth-curve with sharp local minima when n is increased which can be exploited when we seek for possible n to be included in M.
I use $\small w_n = \{h_n\} \cdot n$ and $\small f(n,\varepsilon)=w_n \cdot n^\varepsilon $ with the condition $\small f(n,\varepsilon)<1 $ as criterion for the inclusion of n into the set M . Of course the cardinality of M is limited by an upper bound $ \small n \le N$ with some N that can numerically be handled. I could manage to use $\small N \approx e^{2000}$ within Pari/GP.

The local minima of $\small A_n$ occur near $\small x_k=e^{k-\gamma}, k \in \mathbb N^+$ and the n to be tested is one of the next integers enclosing $\small x_k$, so this must be empirically be determined. (Note that my n are the n+1 used in the comments at the OP denoting the previous high value of A )

The harmonic numbers can be computed in Pari/GP using h(n) = psi(1+n) + Euler ; however, this seems to be limited to something like $ \small n \lt e^{600}$ and so I had to introduce the Euler-McLaurin-formula for the larger $\small n \gt 1e50$

The following table shows the first 40 entries of my 2000-row table using Pari/GP in a couple of seconds.

The basic option is now to test $\small w_n \cdot n^\varepsilon \lt 1$, for an example with $\small \varepsilon=0.1$ see the 6'th column and the 7'th column allowing to sum to the cardinality of M where we find $\small | M(0.1,N) | = 7$ with $\small n \le N \approx e^{2000}$
A second, much nicer, option is to define the function $\small e(n) = -\log_n(w_n) $ and test $\small e(n)>\varepsilon$ whether to include this n or not. This is simply possible looking into the 5'th column. This latter method allows to compute the cardinalities of $\small M(\varepsilon)$ for arbitrary $\varepsilon$ really fast, for instance to create informative scatter- or lineplots.

                n | h_n    | A=frac(h_n) |   w=A*n  |    e(n)   | w*n^0.1|in M?
 -----------------+--------+-------------+----------+-----------+--------+---
                 1  1.00000   1.00000       1.00000     1.000000   1.00000  1
                 4  2.08333   0.0833333     0.333333    0.792481  0.382899  1
                11  3.01988   0.0198773     0.218651    0.634006  0.277901  1
                31  4.02725   0.0272452     0.844601   0.0491822   1.19066  .
                83  5.00207   0.00206827    0.171667    0.398793  0.267051  1
               227  6.00437   0.00436671    0.991243  0.00162136   1.70523  .
               616  7.00127   0.00127410    0.784844   0.0377178   1.49191  .
              1674  8.00049   0.000485572   0.812848   0.0279149   1.70759  .
              4550  9.00021   0.000208063   0.946686  0.00650460   2.19790  .
             12367  10.0000   0.0000430083  0.531883   0.0670005   1.36472  .
             33617  11.0000   0.0000177086  0.595311   0.0497632   1.68811  .
             91380  12.0000   0.00000305167 0.278861    0.111798  0.873923  1
            248397  13.0000   0.00000122948 0.305399   0.0954806   1.05775  .
            675214  14.0000   1.36205E-7    0.919678  0.00623806   3.52030  .
           1835421  15.0000   3.78268E-7    0.694281   0.0252988   2.93703  .
           4989191  16.0000   9.54538E-7    0.476237   0.0481002   2.22652  .
          13562027  17.0000   1.48499E-8    0.201395   0.0975770   1.04059  .
          36865412  18.0000   3.71993E-9    0.137137    0.114033  0.783098  1
         100210581  19.0000   9.73330E-9    0.975380  0.00135314   6.15552  .
         272400600  20.0000   1.61744E-9    0.440592   0.0421997   3.07297  .
         740461601  21.0000   4.01333E-9    0.297172   0.0594162   2.29065  .
        2012783315  22.0000   1.38447E-10   0.278664   0.0596444   2.37389  .
        5471312310  23.0000   1.97920E-11   0.108288   0.0991384   1.01951  .
       14872568831  24.0000   2.27220E-11   0.337935   0.0463183   3.51618  .
       40427833596  25.0000   6.07937E-12   0.245776   0.0574601   2.82623  .
      109894245429  26.0000   7.60776E-12   0.836049  0.00704359   10.6250  .
      298723530401  27.0000   1.82203E-12   0.544283   0.0230213   7.64454  .
      812014744422  28.0000   5.52830E-13   0.448906   0.0292071   6.96806  .
     2207284924203  29.0000   1.00870E-13   0.222650   0.0528504   3.81951  .
     6000022499693  30.0000   2.16954E-14   0.130173   0.0692963   2.46795  .
    16309752131262  31.0000   3.65111E-14   0.595487   0.0170391   12.4772  .
    44334502845080  32.0000   1.81005E-15   0.080247   0.0802804   1.85827  .
   120513673457548  33.0000   4.59281E-15   0.553496   0.0182434   14.1651  .
   327590128640500  34.0000   2.31992E-15   0.759983  0.00821173   21.4950  .
   890482293866031  35.0000   2.71425E-17   0.024169    0.108145  0.755506  1
  2420581837980561  36.0000   2.55560E-16   0.618603   0.0135588   21.3700  .
  6579823624480555  37.0000   1.20561E-17   0.079326   0.0695767   3.02860  .
 17885814992891026  38.0000   4.26642E-18   0.076308   0.0687541   3.21976  .
 48618685882356024  39.0000   7.23781E-19   0.035189   0.0871101   1.64093  .
132159290357566703  40.0000   2.02186E-18   0.267208   0.0334763   13.7708  .

The set $M(0.1)$ is here the set of all $n$ where $\small w_n \cdot n^{0.1} \lt 1$ . The number of such entries for $\small n=132159290357566703 \approx 10^{17}$ is here 7 .
It might be interesting to look at logarithms of the included n . They are

    ln(n)         ln(n) + gamma
  -------------------------------
  0.0       0.57721
  1.38629   1.96351
  2.39789   2.97511
  4.41884   4.99605
  11.4227   12.0000
  17.4227   18.0000
  34.4227   35.0000

This - and the differences - might be interesting for extrapolating a tendency and a guess for the cardinality of $\small M(\varepsilon)$ 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.

# Pari/GP toolsThis

Using the functions

  h(n) = Euler+if(n<1e50, psi(1+n),log(n)+1/2/n-1/12/n^2+1/120/n^4-1/252/n^6+1/240/n^8) 
  A(n) = frac(h(n))
  w(n) = A(n)*n 
  e(n) = if(n==1,return(1)); -log(w(n))/log(n)

The following needs only two steps to find the n with the next local minimum when called with index $\small k \in \mathbb N$ :

 {find_n(k)=local(n1,n2,w1,w2);
   n1=floor(exp(k-Euler)); n2=n1+1; w1 = A(n1);w2 = A(n2);
   if(w1<w2,return(n1),return(n2)); 
    }               
         


 \\ create list --------------------
 {makeList(listlen=40)= local(list,n,w1,logn);
   list=matrix(listlen,3);
   list[1,]=[Euler,1,0,1];
   for(k=2,listlen,
       n=find_n(k); logn=log(n);
       list[k,]=[logn+Euler,n,w1=w(n),-log(w1)/logn];
      );
    return(list); }
    
   \\ compute cardinality with some eps
  cardM(eps,list) = sum(k=1,#list[,1],list[k,4]>eps)     

   \\ apply, note: for long lists we need high internal precision
   list = makeList(40)
   print(cardM(0.1,list) )

(Sorry, I had changed the notation of $\small w(n)$ to $\small f(n)$ in that images and formulae) 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 $\small f(n) = frac(h_n) \cdot n^1$ at the n, where $\small A(n) $ has a local minimum (on request of @GerhardPaseman):

It is also convenient to show a rescaling of the $\small f(n)$ 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 $\small e(n) = - \ln_n(f(n)) \gt \varepsilon$

added image conc. uniformity

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