In[]:=
$Version
Out[]=
14.2.0 for Microsoft Windows (64-bit) (December 26, 2024)
In[]:=
data=Table[{x,PrimePi[x]},{x,40000000,50000000}];
nlm=NonlinearModelFitdata,
x
Loga+
b
Log[x]
,x
,{a,b},x(*
xLog2.682537258154354`+
3.8200129368002025`
Log[x]

Log[x]
*)
Out[]=
FittedModel
xLog2.68+
3.82
Log[x]

Log[x]
Data not saved. Save now

In[]:=
nlm//Normal
Out[]=
xLog2.68254+
3.82001
Log[x]

Log[x]
(*----------------------------------------------------------------------------------*)
In[]:=
P1[x_]:=
xLogE+
2.93
Log[x]

Log[x]
;
In[]:=
GridPrependTable10^n,PrimePi[10^n],DecimalForm[P1[10^n]],DecimalForm[PrimePi[10^n]-P1[10^n]],Abs
(PrimePi[10^n]-P1[10^n])
PrimePi[10^n]
*100,{n,3,19},{"x","RealPi[x]","New Forumla[x]","New Δ","New Δ%"},Frame->All,Alignment{Right}
Out[]=
x
RealPi[x]
New Forumla[x]
New Δ
New Δ%
1000
168
165.756
2.24421
1.33584
10000
1229
1205.9
23.1016
1.87971
100000
9592
9463.25
128.749
1.34226
1000000
78498
77820.2
677.78
0.863436
10000000
664579
660582.
3996.51
0.601359
100000000
5761455
5737394.
24061.1
0.417622
1000000000
50847534
50701749.
145785.
0.286711
10000000000
455052511
454163186.
889325.
0.195434
100000000000
4118054813
4112673052.
5381761.
0.130687
1000000000000
37607912018
37576184907.
31727111.
0.0843629
10000000000000
346065536839
345890858208.
174678631.
0.0504756
100000000000000
3204941750802
3204132547207.
809203595.
0.0252486
1000000000000000
29844570422669
29842717542651.
1852880018.
0.00620843
10000000000000000
279238341033925
279261580879336.
-23239845411.
0.00832258
100000000000000000
2623557157654233
2624069023702171.
-511866047939.
0.0195104
1000000000000000000
24739954287740860
24746926167732500.
-6971879991648.
0.0281807
10000000000000000000
234057667276344607
234139421884727300.
-81754608382656.
0.0349293
In[]:=
P2[x_]:=
xLog2.682537258154354`+
3.8200129368002025`
Log[x]

Log[x]
;
In[]:=
GridPrependTable10^n,PrimePi[10^n],DecimalForm[P2[10^n]],DecimalForm[PrimePi[10^n]-P2[10^n]],Abs
(PrimePi[10^n]-P2[10^n])
PrimePi[10^n]
*100,{n,3,19},{"x","RealPi[x]","My Forumla[x]"," My New Δ","My New Δ%"},Frame->All,Alignment{Right}
Out[]=
x
RealPi[x]
My Forumla[x]
My New Δ
My New Δ%
1000
168
169.982
-1.98229
1.17994
10000
1229
1227.45
1.54536
0.125741
100000
9592
9583.84
8.15709
0.0850406
1000000
78498
78525.1
-27.1152
0.0345426
10000000
664579
664735.
-155.637
0.0234189
100000000
5761455
5761061.
394.311
0.00684394
1000000000
50847534
50823141.
24393.2
0.0479733
10000000000
455052511
454606818.
445693.
0.0979431
100000000000
4118054813
4111829149.
6225664.
0.15118
1000000000000
37607912018
37530872368.
77039650.
0.20485
10000000000000
346065536839
345176950757.
888586082.
0.256768
100000000000000
3204941750802
3195135731249.
9806019553.
0.305966
1000000000000000
29844570422669
29739492482034.
105077940635.
0.352084
10000000000000000
279238341033925
278135156051268.
1103184982657.
0.395069
100000000000000000
2623557157654233
2612144100223473.
11413057430760.
0.435022
1000000000000000000
24739954287740860
24623152228908450.
116802058832412.
0.472119
10000000000000000000
234057667276344607
232872008156208500.
1185659120136128.
0.506567