Somewhat likely looking constructions for the MRB constant

NIntegrate
1/x
x
-1
Sin[Pix]
,{x,1,-InfinityI}//Im
Out[]=
0.18786
​
NIntegrate
1/x
x
-1
Sin[Pix]
,{x,1-1/10^2,-InfinityI}//Im
Out[]=
0.18786
NIntegrate
1/x
x
-1
Sin[Pix]
,{x,1/2,-InfinityI}//Im
Out[]=
0.18786
In[]:=
NIntegrate
1/x
x
-1
Sin[Pix]
,{x,1/10^2,-InfinityI}//Im
NIntegrate
:NIntegrate failed to converge to prescribed accuracy after 9 recursive bisections in x near {x} = {0.01-0.0666843}. NIntegrate obtained -1.63854+0.18786 and 0.0000897467 for the integral and error estimates.
Out[]=
0.18786
​
NIntegrate
1/x
x
-1
Cos[Pi(x+1/2)]
,{x,1,InfinityI}//Im
Out[]=
0.18786
NIntegrate
1/x
x
-1
Cos[Pi(x+1/2)]
,{x,1/2,InfinityI}//Im
Out[]=
0.18786
NSum[Im[
x+1/2
(-1)
(
1/x
x
-1)],{x,1,Infinity}]
Out[]=
0.18786
NSum
1/x
x
-1
Sin[Pi(x+1/2)]
,{x,1,Infinity}
Out[]=
0.18786
NSum[Re[
x
(-1)
(
1/x
x
-1)],{x,1,Infinity}]
Out[]=
0.18786
NSum
1/x
x
-1
Cos[Pix]
,{x,1,Infinity}
Out[]=
0.18786
In[]:=
​​​​
Notice two curios of the vertical distances between the partial sums and integrations and their limit-points:

m is approximated by exactly 10 partial sums in special ways. Why 10?

f[x_]=(-1)^xf[x]
Out[]=
x
(-1)
1
x
x
{m,1/2-Sum[f[k],{k,1,10}]}//N
Out[]=
{0.18786,0.186768}
{-m,Sum[Mean[{
f[3k],f[5k],f[7k]
}],{k,1,10}]}//N
Out[]=
{-0.18786,-0.183419}
In[]:=
f[x_]=(-1)^x
1
x
x
-1;
{m,1/2-Sum[Total[{f[2k],3f[3k],3f[3k],f[4k]}]/11,{k,1,10}]}//N
Out[]=
{0.18786,0.187378}
WolframAlpha["What is the Integral Test?",{{"DefinitionPod:CalculusResult",1},"ComputableData"}]
Out[]=
{Let f(x) be a real-valued function that is continuous and decreasing in [1, infinity) and satisfies f(x)>=0 for all x in [1, infinity). Set a_n = f(n).,1.If integral_1^infinity f(x) dx is convergent, then sum_(n=1)^infinity a_n is convergent.,2.If integral_1^infinity f(x) dx is divergent, then sum_(n=1)^infinity a_n is divergent.,In fact, the test can be stated for any integral [N, infinity), where N is a natural number, instead of [1, infinity).}
Forf(x)=(
1/x
x
-1)*Cos[Pi*x]
​

The integrated analog of m,
∞
∫
1
-1+
1
x
x
Cos[πx]x
, is similarly approximated below.

(Sum[Mean[{f[k],
f[3k]
,
f[5k]
,
f[7k]
,f[9k]}],{k,1,10}]-7)//RealDigits[#,10,10]&//First,NIntegrate[(
1/x
x
-1)*Cos[Pi*x],{x,1,Infinity},WorkingPrecision10]//RealDigits[#,10,10]&//First
NIntegrate
:DoubleExponentialOscillatory returns a finite integral estimate, but the integral might be divergent.
Out[]=
{{7,0,7,7,4,7,7,7,0,0},{7,0,7,7,6,0,3,9,4,0}}

Somewhat likely looking constructions for the MRB constant’s integrated analog

NIntegrate
1/x
x
-1
ISin[-Pix]+Cos[-Pix]
,{x,1,InfinityI}
Out[]=
0.070776-0.0473806
NIntegrate
1/x
x
-1
Exp[-PixI]
,{x,1,InfinityI}
Out[]=
0.070776-0.0473806
NIntegrate[Exp[PixI](
1/x
x
-1),{x,1,InfinityI}]
Out[]=
0.070776-0.0473806
)

Unlikely looking possible construction for the MRB constant’s integrated analog

​​NIntegrateE^(IPix)
1
x
x
-1,{x,1,InfinityI},WorkingPrecision9
Out[]=
0.0707760383-0.0473806179
mkb=NIntegrateE^(IPix)
1
x
x
-1,{x,1,InfinityI},WorkingPrecision20
Out[]=
0.070776039311528803669-0.047380617070350786074

What about
== the MRB constant analog?

In[]:=
f[n_]:=MeijerG[{{},Table[1,{n+1}]},{Prepend[Table[0,n+1],-n+1],{}},-π];
Table[Print["to ",x," terms gives an error of ",mkb-Sum[N[(I/Pi)^(1-n)*f[n],x],{n,1,x}]],{x,2,12}];
to 2 terms gives an error of 0.001+0.×
-4
10

to 3 terms gives an error of 0.0001+0.×
-5
10

to 4 terms gives an error of 0.×
-6
10
+0.×
-6
10

to 5 terms gives an error of 0.×
-7
10
+0.×
-7
10
