Some relationships between the MKB and the MRB constant

f(x)=
x
(-1)
1-
1
x+1
(x+1)

h(x)=
x
(-1)
(
1/x
x
-1)
g(x)=
1/x
x
CMRB=
∞
∑
n=0
f(n)=
∞
∑
n=1
f(n)=
∞
∑
n=1
h(n)
MKB=

N∞
2N+1
∫
1
πt

g(t)t=-
∞
∫
0
g(1+t)
πt

t+

π

∞
∫
0
((f(t)-f(-t))/(
2πt

-1))t+

N∞
2N+1
∫
1
πt

g(t)t=CMRB
In[]:=
f(x_)=
x
(-1)
1-
1
x+1
(x+1)
;h(x_)=
x
(-1)
(
1/x
x
-1);g(x_)=
1/x
x
;
In[]:=
NIntegrate[E^(Pi*I*t)*g[t],{t,1,10001},WorkingPrecision->10]
Out[]=
0.07079022316-0.04706930965
In[]:=
MKB=-INIntegrate[g[1+It]/E^(Pit),{t,0,Infinity},WorkingPrecision100,MaxRecursion20]-I/Pi
Out[]=
0.0707760393115288035395280218302820013657546962033630275831727881636184572643820365808318812661772382-0.6840003894379321291827444599926611267109914826549994343226303771381530581249766381509598342127214787
In[]:=
CMRB=NSum[h[n],{n,1,Infinity},Method"AlternatingSigns",WorkingPrecision100];
In[]:=
term=NIntegrate[((I(f[It]-f[-It])))/(E^(2Pit)-1),{t,0,Infinity},WorkingPrecision100,Method"GlobalAdaptive"]
Out[]=
0.11708360315053831670898991222399122869014839869677575858883189592585877430027817712246477316693025869+0.04738061707035078610720940650260367857315289969317363933196100090256586758807049779050462314770913485
In[]:=
(term+MKB-CMRB)
Out[]=
9.3472×
-94
10
+0.×
-101
10

In[]:=
NoMKB=I
∞
∫
0
h(1-t)-h(1+t)
exp(2πt)-1
t.
In[]:=
NoMKB=Im[NIntegrate[(h[(1-It)]-h[(1+It)])/(Exp[2Pit]-1),{t,​​0,Infinity},WorkingPrecision100]]
Out[]=
0.11708360315053831670898991222399122869014839869677575858883189592585877430027817712246477316693025869
In[]:=
NoMKB+Re[MKB]-CMRB
Out[]=
9.3472×
-94
10
In[]:=
term+2I/Pi
Out[]=
0.11708360315053831670898991222399122869014839869677575858883189592585877430027817712246477316693025869+0.68400038943793212918274445999266112671099148265499943432263037713815305812497663815095983421272147867
∞
∫
0
(h(1+t)-h(1-t))
2πt

-1
t-
2
π
=
∞
∫
0
(f(t)-f(-t))
2πt

-1
In[]:=
-NIntegrate[((I(h[1+It]-h[1-It])))/(E^(2Pit)-1),{t,0,Infinity},WorkingPrecision100,Method"GlobalAdaptive"]-2I/Pi
Out[]=
-0.11708360315053831670898991222399122869014839869677575858883189592585877430027817712246477316693025869-0.68400038943793212918274445999266112671099148265499943432263037713815305812497663815095983421272147867
In[]:=
NoMKB+Im[MKB]I
Out[]=
0.11708360315053831670898991222399122869014839869677575858883189592585877430027817712246477316693025869-0.6840003894379321291827444599926611267109914826549994343226303771381530581249766381509598342127214787

∞
∫
0
(h(1+t)-h(1-t))
2πt

-1
t==
∞
∫
0
(f(t)-f(-t))
2πt

-1
==term
In[]:=
term-NIntegrate[((I(h[1+It]-h[1-It])))/(E^(2Pit)-1),{t,0,Infinity},WorkingPrecision100,Method"GlobalAdaptive"]
Out[]=
0.×
-101
10
+0.×
-101
10

In[]:=
NIntegrate[((I(f[It]-f[-It])))/(E^(2Pit)-1),{t,0,Infinity},WorkingPrecision100,Method"GlobalAdaptive"]-term
Out[]=
0.×
-101
10
+0.×
-101
10

In[]:=
MKB
Out[]=
0.0707760393115288035395280218302820013657546962033630275831727881636184572643820365808318812661772382-0.6840003894379321291827444599926611267109914826549994343226303771381530581249766381509598342127214787