Using this
,
we get the following for the MRB constant (CMRB).
f(x_)=
1/x
x
-1;g(x_)=exp(πx)f(x);
​​
CMRB=
∞
∑
n=1
g(n)
then
g1(x_)=
1/x
x
;
​​
CMRB=
1
2

∞
∫
0
g1(1-t)-g1(t+1)
sin(πt)
t
then finally,
f1
(
x_
)
=
1-
1/x
x
;
g2(x_)=
f1(1-x)
exp(πx)-exp(-πx)
;
u:=1-t
;
​​
CMRB=2
1
∫
0
Img2
t
u

2
u
t.
​​​​
Using this
we get the following, where m is the MRB constant (CMRB).
f(x_)=
1/x
x
-1;g(x_)=exp(πx)f(x);
​​
m=
∞
∑
n=1
g(n)
In[]:=
f[x_]=-1+x^(1/x);g[x_]=f[x]Exp[PiIx];m=N[NSum[g[n],{n,1,Infinity},Method"AlternatingSigns",WorkingPrecision107],100]
Out[]=
0.1878596424620671202485179340542732300559030949001387861720046840894772315646602137032966544331074969
gs(x_)=
1/x
x
;
​​
m=
1
2

∞
∫
0
gs(1-t)-gs(t+1)
sin(πt)
t
In[]:=
gs[x_]=x^(1/x);​​m-I/2NIntegrate[(gs[(1-t)]-gs[(1+t)])/(Sin[πt]),{t,0,InfinityI},WorkingPrecision100]
Out[]=
0.×
-101
10
fp(x_)=1-
1/x
x
;gt(x_)=
fp(1-x)
exp(πx)-exp(-πx)
;u:=1-t;
​​
m=2
1
∫
0
Imgt
t
u

2
u
t
In[]:=
​​fp[x_]=1-x^(1/x);gt[x_]=(fp[(-xI+1)])/(Exp[Pix]-Exp[-Pix]);u:=1-t;m-2NIntegrate[Im[gt[t/u]]/u^2,{t,0,1},WorkingPrecision100]
Out[]=
0.×
-101
10