20,000 digits of the MKB constant (M1)

Marvin Ray Burns

Abstract.

We use a combination of four acceleration methods here. Using formula (12) in1 (http : // arxiv.org/pdf/0912.3844 v3.pdf), the 2NIntegrate command with AccuracyGoal option of Mathematica 10.1, simplifying the integrand and breaking the oscillating integral into real and imaginary parts:  we compute the real part, the digits of the absolute value of the imaginary part, and the magnitude of the MKB constant; with timing of each. So the real and imaginary parts are computed separately as positive, real constants.

1.  Definition

MKB constant = M1=(Abs[(lim_(N->∞))(∫_1^(2 N))(e^(i π x))(x^(1/x)) dx])
Since the MKB constant is the integral analog of the MRB constant, it is named after the MRB constant’s author’s first wife.

2.  Numerical Analysis

The specific algorithm used, by applying partial integration up to the 30th derivative, (as shown in http://arxiv.org/pdf/0912.3844v3.pdf ), breaking into real and imaginary parts for easier integration, we get

d=30;f(x_)=(x^(1/x));
a(n_):=(∑_(k=1)^n)(StirlingS1[n, k]) (∑_(j=0)^k)(((-j))^(k-j)) (Binomial[k, j]);a(0)=1;g=((2 I)/π)-(∑_(n=1)^d)-(((I^(n+1)) a(n))/(π^(n+1)));
a=((((1/π)))^d) (-(∫_1^∞)cos(π x) (((∂^d)f(x))/(∂(x^d)))\[DifferentialD]x)-Re(g)
b=((((1/π)))^(d+1)) ((∫_1^∞)cos(π x) (((∂^(d+1))f(x))/(∂(x^(d+1))))\[DifferentialD]x)+Im(g)
M1=(Sqrt[(a^2)+(b^2)])

In the following program the Simplify command is needed for the integral to “converge” in Mathematica 10.1.

3.  The Program and Results

This program can be downloaded from  http://marvinrayburns.com/20KMKB.txt .

d=30;f[x_]=x^(1/x);ClearAll[a,b,h];
a[n_]:=Sum[StirlingS1[n,k]*Sum[(-j)^(k-j)*Binomial[k,j],{j,0,k}],{k,1,n}];a[0]=1;g=2 I/Pi-Sum[-I^(n+1) a[n]/Pi^(n+1),{n,1,d}];Print[DateString[]];
Print[T0=SessionTime[]];prec=20000;
Print[N[a=-Re[g]-(1/Pi)^(d)*NIntegrate[Simplify[Cos[Pi*x]*D[f[x],{x,d}]],{x,1,Infinity},WorkingPrecision->prec*(105/100),PrecisionGoal->prec*(105/100)],prec]];
Print[SessionTime[]-T0," seconds"];
Print[N[b=Im[g]+(1/Pi)^(d+1)*NIntegrate[Simplify[Cos[Pi*x]*D[f[x],{x,d+1}]],{x,1,Infinity},WorkingPrecision->prec*(105/100),PrecisionGoal->prec*(105/100)],prec]];
Print[SessionTime[]-T0," seconds"];Print[N[Sqrt[a^2+b^2],prec]];Print[DateString[]];

4. Summary

The real part of M1 is 0.070776039311528803539... . It is important to note that the imaginary part (above displayed as a positive, real constant) is actually -0.684000389437932129182...I .Each part took about 17 hours to calculate, These digits have been checked with a 25,000 digit run, completed on Fri 15 May 2015 11:52:20,; they all were correct! The program for 25,000 digits follows/:
d = 35; f[x_] = x^(1/x); ClearAll[a, b, h];
h[n_] := Sum[
  StirlingS1[n, k]*Sum[(-j)^(k - j)*Binomial[k, j], {j, 0, k}], {k, 1,
    n}]; h[0] = 1; g = 
 2 I/Pi - Sum[-I^(n + 1) h[n]/Pi^(n + 1), {n, 1, d}]; Print[
 DateString[]];
Print[T0 = SessionTime[]]; prec = 25000;
Print[N[a = -Re[g] + (1/Pi)^(d + 1)*
      NIntegrate[
       Simplify[Cos[Pi*x]*D[f[x], {x, d + 1}]], {x, 1, Infinity}, 
       WorkingPrecision -> prec*(105/100), 
       PrecisionGoal -> prec*(105/100)], prec]];
Print[SessionTime[] - T0, “ seconds”];
Print[N[b = 
    Im[g] - (1/Pi)^(d + 1)*
      NIntegrate[
       Simplify[Sin[Pi*x]*D[f[x], {x, d + 1}]], {x, 1, Infinity}, 
       WorkingPrecision -> prec*(105/100), 
       PrecisionGoal -> prec*(105/100)], prec]];
Print[SessionTime[] - T0, “ seconds”]; Print[
 N[Sqrt[a^2 + b^2], prec]]; Print[DateString[]];
Other records follow.

Here is the algorithm for the 25,000 digit run:

d=35;f(x_)=(x^(1/x));
h(n_):=(∑_(k=1)^n)(StirlingS1[n, k]) (∑_(j=0)^k)(((-j))^(k-j)) (Binomial[k, j]);h(0)=1;g=((2 i)/π)-(∑_(n=1)^d)-(((i^(n+1)) h(n))/(π^(n+1)));
a=((((1/π)))^(d+1)) (∫_1^∞)cos(π x) (((∂^(d+1))f(x))/(∂(x^(d+1))))dx-Re(g);
b=Im(g)-((((1/π)))^(d+1)) (∫_1^∞)sin(π x) (((∂^(d+1))f(x))/(∂(x^(d+1))))dx;
M1=(Sqrt[(a^2)+(b^2)])

A raw mathematica notbook is available with the program and results of the 25,000 digit run at http://www.marvinrayburns.com/MKB25K1.nb.

5. Table

Table 1
"This table does not reflect all of the work 
Richard Mathar did on this constant and the number of digits of his records are just an estimate"

These records make for the following graph, drawn by Felis Phasma.

6. Graph

Graph 1

7. Open Questions

We ask two questions: Is there a simpler algorithm that effectively computes 20000 digits of the MKB constant?And can it be done faster? A faster progam is presntly being tested.

8. References

1	http : // arxiv.org/pdf/0912.3844 v3.pdf

2	https://reference.wolfram.com/language/ref/AccuracyGoal.html