In[]:=
(*MRBConstant:Borel-LaplaceandHurwitz-ZetaAcceleration*)(*MarvinRayBurns-October2026*)Module[{mMax=60,kSub=4,kLast=8,u,s,z,poly,c,b,g,borel,lead,raw,correction,result,reference},(*ExactrationalTaylorcoefficients*)poly=Normal[Series[Exp[Log[1+u]/(1+u)]-1,{u,0,2kLast+3}]];c[k_Integer?NonNegative]:=(-1)^kCoefficient[poly,u,2k+1];(*RationalBernoulli-Borelcoefficients*)b[k_Integer?NonNegative]:=(-1)^kc[k](2^(2k+2)-1)*BernoulliB[2k+2]/Factorial[2k+2];g[z_?NumericQ]:=Exp[Log[z]/z]-1;(*AcceleratedBoreltransform*)borel[z_?NumericQ]:=Sum[b[k]z^(2k+1),{k,0,kSub}]+(2/Pi)Sum[With[{y=z/(Pi(2j+1))},(Im[g[1+Iy]]-Sum[c[k]y^(2k+1),{k,0,kSub}])/(2j+1)],{j,0,mMax}];(*Hurwitz-zetatailcorrections*)lead[k_Integer?NonNegative]:=c[k+1]Factorial[2k+3]*Zeta[2k+4,mMax+3/2]/(2^(2k+3)Pi^(2k+4));(*Borel-Laplaceintegration*)raw=NIntegrate[Exp[-s]borel[s],{s,0,Infinity},WorkingPrecision->75,AccuracyGoal->43,PrecisionGoal->43];(*Fiveanalyticcorrections*)correction=N[Sum[lead[k],{k,kSub,kLast}],65];result=raw+correction;(*Referencevalue,storedasanexactrational*)reference=187859642462067120248517934054273230055903094900138786/10^54;Print[Style["MRB Constant: Borel-Laplace Evaluation",16,Bold]];Grid[{{"Borel-Laplace integral",N[raw,40]},{"Hurwitz-zeta correction",N[correction,16]},{"Corrected MRB constant",N[result,40]},{"Reference MRB constant",N[reference,40]},{"Difference",N[result-reference,12]}},Frame->All,Alignment->Left,Spacings->{2,1}]]
MRB Constant: Borel-Laplace Evaluation
Out[]=
Borel-Laplace integral | 0.1878596424620671202485543897174549037329 |
Hurwitz-zeta correction | -3.645566318167268× -23 10 |
Corrected MRB constant | 0.1878596424620671202485179340542732310540 |
Reference MRB constant | 0.1878596424620671202485179340542732300559 |
Difference | 9.98127667865× -37 10 |