10,000 digits of
$nul = -0.0760867...,$ the hard to calculate nul, not necessarily
$ \sum _{m=2}^{\infty } (-1)^m \left( \eta ^{(m)}(m)\right)=-0.0724829...?.$
m = NSum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity},
Method -> "AlternatingSigns", WorkingPrecision -> 10000];
sum = 1/2 Log[2] (-2 EulerGamma + Log[2]):
nul = (-m - sum)*E
-0.07608671642673194446000...
Here is how to try compute it without computing the MRB constant.
First compute MRBeta2toinf=
$\sum _{m=2}^{\infty } (-1)^m \frac{\left( \eta ^{(m)}(m)\right)}{m!}.$
That is the Cradall first eta formula for MRB -sum.
Print["Start time is " "Start time is ", ds = DateString[], "."];
prec = 10000;
(**Number of required decimals.*.*)ClearSystemCache[];
T0 = SessionTime[];
expM[pre_] :=
Module[{lg, a, d, s, k, bb, c, end, iprec, xvals, x, pc, cores = 16(*=
4*number of physical cores*), tsize = 2^7, chunksize, start = 1,
ll, ctab, pr = Floor[1.0002 pre]}, chunksize = cores*tsize;
n = Floor[1.32 pr];
end = Ceiling[n/chunksize];
Print["Iterations required: ", n];
Print["Will give ", end,
" time estimates, each more accurate than the previous."];
Print["Will stop at ", end*chunksize,
" iterations to ensure precsion of around ", pr,
" decimal places."]; d = ChebyshevT[n, 3];
{b, c, s} = {SetPrecision[-1, 1.1*n], -d, 0};
iprec = pr/2^6;
Do[xvals = Flatten[ParallelTable[Table[ll = start + j*tsize + l;
lg = Log[ll]/(ll); x = N[E^(lg), iprec];
pc = iprec;
While[pc < pr, pc = Min[4 pc, pr];
x = SetPrecision[x, pc];
xll = x^ll; z = (ll - xll)/xll;
t = 2 ll - 1; t2 = t^2;
x =
x*(1 + SetPrecision[4.5, pc] (ll - 1)/
t2 + (ll + 1) z/(2 ll t) -
SetPrecision[13.5, 2 pc] ll (ll - 1)/(3 ll t2 + t^3 z))];
x - lg, {l, 0, tsize - 1}], {j, 0, cores - 1},
Method -> "EvaluationsPerKernel" -> 16]];
ctab = ParallelTable[Table[c = b - c;
ll = start + l - 2;
b *= 2 (ll + n) (ll - n)/((ll + 1) (2 ll + 1));
c, {l, chunksize}], Method -> "EvaluationsPerKernel" -> 16];
s += ctab.(xvals - 1);
start += chunksize;
st = SessionTime[] - T0; kc = k*chunksize;
ti = (st)/(kc + 10^-10)*(n)/(3600)/(24);
If[kc > 1,
Print[kc, " iterations done in ", N[st - stt, 4], " seconds.",
" Should take ", N[ti, 4], " days or ", ti*3600*24,
"s, finish ", DatePlus[ds, ti], "."],
Print["Denominator computed in ", stt = st, "s."]];, {k, 0,
end - 1}];
N[-s/d, pr]];
t2 = Timing[MRBeta2toinf = expM[prec];]; Print["Finished on ",
DateString[], ". Processor and total time were ",
t2[[1]], " and ", st, " s respectively."];
Then
-(MRBeta2toinf*E)
-0.07608671642673194446000...