f(t_)=
1
1+t
(1+t)
;​​TheMRBconstant​​=m=​​
∞
∫
0
Im
f(t)
sinh(πt)
t
​
(*Overthespecifiedrange,*)-
t
π
+
Im
f(t)
sinh(πt)
Re
f(t)
sinh(πt)
t-
-2
(10Log[6])
=4.71902*
-7
10
​​
In[]:=
f[t_]=(1+It)^(1/(1+It));
In[]:=
m=NIntegrate[Im[f[t]/Sinh[Pit]],{t,0,Infinity},WorkingPrecision->18]
Out[]=
0.187859642462067159
In[]:=
b=t/.FindRoot[E/Pit==Im[f[t]/Sinh[Pit]]/Re[f[t]/Sinh[Pit]],{t,1,Pi},WorkingPrecision->30](*(0,b)isthespecifiedrangeoftheaboveandfollowingintegral.*)
Out[]=
0.306741782076160452661440435549
In[]:=
int=NIntegrate[-E/Pit+Im[f[t]/Sinh[Pit]]/Re[f[t]/Sinh[Pit]],{t,0,b}]
Out[]=
0.00311535
In[]:=
d=
-2
(10Log[6])
;d//N
Out[]=
0.00311487
In[]:=
int-d
Out[]=
4.71902×
-7
10
In[]:=
m-
21
20
+

π
-d
Out[]=
7.47179637988×
-7
10
In[]:=
5647
100
4
10
E*m
∑
i=1
p
i
-int+d
Out[]=
-9.68652×
-12
10
In[]:=
1
4
10
E*m
∑
i=1
p
i
Out[]=
1
119667031
In[]:=
5647
100.
%
Out[]=
4.71893×
-7
10
In[]:=
-int+d
Out[]=
-4.71902×
-7
10
In[]:=
N1-
1
4
10
E*m
∑
i=1
p
i
,20
Out[]=
0.99999999164347948099
In[]:=
m-
E
Pi
-
7
5
AGM1,
2

Out[]=
0.999999991659631162
Using them, we find the following equations.
​​TheMRBconstant​​=m=
∞
∑
n=1
n
(-1)

n
n
-1..​​mshort==
1256012054
∑
n=1
n
(-1)

n
n
-1​​
p
i
isthei'thprime.
m-1+

π
-
-2
10
m
=1.1560898070373*
-4
10
​​
m-1-

π
-
7
5
AGM[1,
2
]==-8.3403688765*
-9
10
​​
mshort-1-

π
-
7
5
AGM[1,
2
]==-7.4188344824*
-18
10
​​m-mshort==-8.3403688691
-9
10
​​
m-mshort+
1
4
10
Em
∑
i=1
p
i
==1.6151649871*
-11
10
​​
While
​​
1/5>m>1/6
,
​
​
Absm-mshort+
1
4
10
Em
∑
i=1
p
i

<
Absm-mshort+
1
4
10
E*1/5
∑
i=1
p
i

<
Absm-mshort+
1
4
10
E*1/6
∑
i=1
p
i

.
​​
In[]:=
m=NSum[
n
(-1)
(
1/n
n
-1),{n,1,Infinity},Method->"AlternatingSigns",WorkingPrecision->60];
In[]:=
mshort=NSum[
n
(-1)
(
1/n
n
-1),{n,1,1256012054},Method->"AlternatingSigns",WorkingPrecision->60];
In[]:=
AGM=ArithmeticGeometricMean;
​
In[]:=
m-1+

π
-
-2
10
m
Out[]=
-0.0001156089807037326228944352013174373056636658505409583881
In[]:=
m-1-

π
AGM[1,
2
]
+
7
5
Out[]=
-6.96109573383447332705750332585149672785921117178085×
-9
10
In[]:=
m-1-

π
-
7
5
AGM[1,
2
]
Out[]=
-8.3403688765533653663993629578547438005064645888310×
-9
10
In[]:=
mshort-1-

π
-
7
5
AGM[1,
2
]
Out[]=
-7.4188344824006461094669008007315074532082×
-18
10
In[]:=
N[m-mshort,20]
Out[]=
-8.3403688691345308840×
-9
10
In[]:=
1/5>m>1/6
Out[]=
True
In[]:=
Absm-mshort+
1
4
10
E*m
∑
i=1
p
i
<Absm-mshort+
1
4
10
E*1/5
∑
i=1
p
i
<Absm-mshort+
1
4
10
E*1/6
∑
i=1
p
i

Out[]=
True
From the above equation,
we have