Examples of further research unveiling the utility of the MRB constant

The MRB constant early interest

As early as 1999, after I emailed him, technical writer at MathSoft Inc. Dr. Steven Finch shared the following query about the MRB constant;

Marvin Ray Burns asked about $$ \lim_{N \to \infty} \sum_{n=1}^{2N} (-1)^n n^{\frac{1}{n}} = \sum_{k=1}^{\infty} \left( (2k)^{\frac{1}{2k}} - (2k-1)^{\frac{1}{2k-1}} \right) $$

which is slowly convergent (on the order of only $\ln(k)/k^2 $). No exact formula for this series is known, although it bears some resemblance to expressions mentioned with the Glaisher-Kinkelin constant and with infinite product constants. It may also be rewritten as

$$ =\sum_{m=1}^{\infty} (-1)^m \left( m^{\frac{1}{m}} - 1 \right) = 1 + \lim_{N \to \infty} \sum_{n=1}^{2N+1} (-1)^n n^{\frac{1}{n}} = 0.1878596424\ldots $$

Using Cohen-Villegas-Zagier's acceleration technique of convergent alternating series, Burns obtained a high-precision estimate of this constant. I am not certain if required hypotheses are met. Can someone help here?

Before his MIT Sloan School of Management's current role as Research Computing Specialist, in 2003 Statistical Computing Scientist/Preceptor at the Harvard University Department of Statistics, Professor Steven Finch was the first to publish about the MRB constant in book form under the auspices of Cambridge Press:

A more difficult evaluation concerns the series

$$ \lim_{N \to \infty} \sum_{n=1}^{2N} (-1)^n n^{\frac{1}{n}} = \sum_{k=1}^{\infty} \left( (2k)^{\frac{1}{2k}} - (2k-1)^{\frac{1}{2k-1}} \right) $$

$$ =\sum_{m=1}^{\infty} (-1)^m \left( m^{\frac{1}{m}} - 1 \right) = 1 + \lim_{N \to \infty} \sum_{n=1}^{2N+1} (-1)^n n^{\frac{1}{n}} = 0.1878596424\ldots $$

which is slowly convergent. No exact formulas are known, although the series bear some resemblance to expressions mentioned in [2.15]. Cesàro summation and Cohen-Villegas-Zagier acceleration are two techniques available to compute the sum.

The MRB constant integrated analog

On February 22 2009, I asked:

it appears that the absolute value, minus 1/2, of the integral of (-1)^x*x^(1/x) from 1 to infinity would equal the partial sum of (-1)^x*x^(1/x) from 1 to where the upper summation is even and growing without bound. Is anyone interested in improving or disproving this conjecture?

As found here, whether in response or for other reasons,

Richard J. Mathar of CERN published the foundational work of its integrated analog.

INTEGRAL OVER exp(ix)x^{1/x} BETWEEN 1 AND INFINITY arXiv:0912.3844v3 [math.CA] 5 Aug 2010

NUMERICAL EVALUATION OF THE OSCILLATORY INTEGRAL OVER $\exp(i \pi x)x^{1/x}$ BETWEEN 1 AND INFINITY

RICHARD J. MATHAR

Real and imaginary part of the limit $2N \to \infty$ of the integral

$$ \int_{1}^{2N} \exp(i\pi x) x^{1/x} dx $$

are evaluated to 20 digits with brute force methods after multiple partial integration, or combining a standard Simpson integration over the first half wave with series acceleration techniques for the alternating series co-phased to each of its points. The integrand is of the logarithmic kind; its branch cut limits the performance of integration techniques that rely on smooth higher order derivatives.

First, following-up on my inquiries about the efficient computation of the MRB constant digits mentioned in the above paper, Henrik Schachner, --from the Radiation Therapy Center in Weilheim, Germany, who is a physicist with a PhD from the University of Regensburg and currently working in medical physics, now in radiotherapy, previously in radiology had the most instructive remark.

here:

ClearAll["Global`*"]
m = Table[N[(-1)^k (k^(1/k) - 1), 2000], {k, 1, 2000}];
(*partial sums of the series*)
am = Accumulate@m;
shanks[ac_List] := 
 Table[(ac[[n + 1]] ac[[n - 1]] - ac[[n]]^2)/(ac[[n + 1]] + 
     ac[[n - 1]] - 2 ac[[n]]), {n, 2, Length[ac] - 1}]
sac = NestList[shanks, am, 24];
ListLogPlot[Abs[Differences /@ sac], Joined -> True, 
 GridLines -> Automatic, ImageSize -> Large]

This plot showcases the Shanks transformation that can be applied iteratively, leading to further improvements in MRB constant calculations.

Studying Mathar R. J. has led to many programs and formulas to compute the integrated analog of the MRB constant.

The efficient programs

Wed 29 Jul 2015 11:40:10

From an initial accuracy of only 7 digits,

0.07077603931152880353952802183028200137`19.163032309866352 - 
 0.68400038943793212918274445999266112671`20.1482024033675 I - \
(NIntegrate[(-1)^t (t^(1/t) - 1), {t, 1, Infinity}, 
    WorkingPrecision -> 20] - 2 I/Pi)

(5.256245460165610^-7 - 5.0557218633576410^-6 I)

we have the first efficient program to compute the integrated analog (MKB) of the MRB constant, which is good for 35,000 digits.

Block[{$MaxExtraPrecision = 200}, prec = 4000; f[x_] = x^(1/x);
 ClearAll[a, b, h];
 Print[DateString[]];
 Print[T0 = SessionTime[]];

 If[prec > 35000, d = Ceiling[0.002 prec], 
  d = Ceiling[0.264086 + 0.00143657 prec]];

 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}];

 sinplus1 := 
  NIntegrate[
   Simplify[Sin[Pi*x]*D[f[x], {x, d + 1}]], {x, 1, Infinity}, 
   WorkingPrecision -> prec*(105/100), 
   PrecisionGoal -> prec*(105/100)];

 cosplus1 := 
  NIntegrate[
   Simplify[Cos[Pi*x]*D[f[x], {x, d + 1}]], {x, 1, Infinity}, 
   WorkingPrecision -> prec*(105/100), 
   PrecisionGoal -> prec*(105/100)];

 middle := Print[SessionTime[] - T0, " seconds"];

 end := Module[{}, Print[SessionTime[] - T0, " seconds"];
   Print[c = Abs[a + b]]; Print[DateString[]]];


 If[Mod[d, 4] == 0, 
  Print[N[a = -Re[g] - (1/Pi)^(d + 1)*sinplus1, prec]];
  middle;
  Print[N[b = -I (Im[g] - (1/Pi)^(d + 1)*cosplus1), prec]];
  end];


 If[Mod[d, 4] == 1, 
  Print[N[a = -Re[g] - (1/Pi)^(d + 1)*cosplus1, prec]];
  middle;
  Print[N[b = -I (Im[g] + (1/Pi)^(d + 1)*sinplus1), prec]]; end];

 If[Mod[d, 4] == 2, 
  Print[N[a = -Re[g] + (1/Pi)^(d + 1)*sinplus1, prec]];
  middle;
  Print[N[b = -I (Im[g] + (1/Pi)^(d + 1)*cosplus1), prec]];
  end];

 If[Mod[d, 4] == 3, 
  Print[N[a = -Re[g] + (1/Pi)^(d + 1)*cosplus1, prec]];
  middle;
  Print[N[b = -I (Im[g] - (1/Pi)^(d + 1)*sinplus1), prec]];
  end];]

May 2018

I got substantial improvement in calculating the digits of MKB by using V11.3 in May 2018, my new computer (processor Intel(R) Core(TM) i7-7700 CPU @ 3.60GHz, 3601 MHz, 4 Core(s), 8 Logical Processor(s) with 16 GB 2400 MH DDR4 RAM):

Digits  Seconds
2000    67.5503022
3000    217.096312
4000    514.48334
5000    1005.936397
10000   8327.18526
 20000  71000

They are found in the attached 2018 quad MKB.nb.

They are twice as fast,(or more) as my old records with the same program using Mathematica 10.2 in July 2015 on my old big computer (a six-core Intel(R) Core(TM) i7-3930K CPU @ 3.20 GHz 3.20 GHz with 64 GB of 1066 MHz DDR3 RAM):

digits          seconds

2000    256.3853590 
3000    794.4361122
4000       1633.5822870
5000        2858.9390025
10000      17678.7446323 
20000      121431.1895170
40000       I got error msg

May 2021

After finding the following rapidly converging integral for MKB, I finally computed 200,000 digits of MKB (0.070776 - 0.684 I...) Started ‎Saturday, ‎May ‎15, ‎2021, ‏‎10: 54: 17 AM, and finished at 9:23:50 am EDT | Friday, August 20, 2021, for a total of 8.37539*10^6 seconds or 96 days 22 hours 29 minutes 50 seconds.

The full computation, verification to 100,000 digits, and hyperlinks to various digits are found below at 200k MKB A.nb. The code was

g[x_] = x^(1/x); u := (t/(1 - t)); Timing[
 MKB1 = (-I Quiet[
      NIntegrate[(g[(1 + u I)])/(Exp[Pi u] (1 - t)^2), {t, 0, 1}, 
       WorkingPrecision -> 200000, Method -> "DoubleExponential", 
       MaxRecursion -> 17]] - I/Pi)]

The Wolfram Notebook assistant proved this.

To construct a detailed proof that demonstrates the equality of the two integrals using series expansions, contour integration, and numerical verification, we can proceed with the following steps: Conclusion Theoretical analysis using series expansions and contour integration, combined with numerical verification, demonstrates the equality of the two integrals. The contour integral approach ensures that the integrals along the real and imaginary axes are equivalent, up to the added term. This illustrates the equivalency of these two complex integrals.

After finding the above more rapidly converging integral for MKB, In only 80.5 days, 189,330 real digits and 166,700 imaginaries were confirmed to be correct by the following different formula. as Seen at https://www.wolframcloud.com/obj/bmmmburns/Published/2nd%20200k%20MRB.nb

All digits at

https://www.wolframcloud.com/obj/bmmmburns/Published/200K%20confirmed%20MKB.nb (Recommended to open in desktop Mathematica.)

N[(Timing[
   FM2200K - (NIntegrate[(Exp[Log[t]/t - Pi t/I]), {t, 1, Infinity I},
        WorkingPrecision -> 200000, Method -> "Trapezoidal", 
       MaxRecursion -> 17] - I/Pi)]), 20]

I've learned more about what MaxRecusion is required for 250,000 digits to be verified from the two different formulas, and they are being computed as I write. It will probably take over 100 days. Let's try to formalize this derivation.

• The MRB constant eta derivations

The continuing development of MRB constant formulas connect the Dirichlet eta to nth roots yielding a proof of the previously proposed concept that the MRB constant connects such ideas together.

Published in his life-time collection, Algorithmic Reflections: Selected Works, professor Richard Crandall of Reed College made the case that the MRB constant is indeed a key fundamental constant

Achieved here

To summarize his thesis.

The MRB constant's eta formula connects various zeta function variants through its use of series and its relationship to other special functions. Let's explore this connection in detail.

The MRB Constant and Its Series Representation The MRB constant (B as he put it) is defined by a specific series involving alternating sums:

This series converges quickly, making it efficient for high-precision calculations. Here's how this ties together various zeta function variants:

Relationship to Zeta Function Variants Eta Function: The eta function η(s) is a variant of the Riemann zeta function ζ(s) and is defined as: $ \eta(s) = (1 - 2^{1-s}) \zeta(s)$

shown here,

This has led to faster convergence of digits, so in search for a suitable approximation to the MRB constant to answer several questions about patterns in its decimal approximation,

From ScienceDirect:

All the formulas for the MRB constant and its integrated analog are in Results from further research unveiling the utility of the MRB constant

PARTIAL SUM METHODS FOR MRB CONSTANT

The success in computing millions of digits of the MRB constant, found in a following reply, lies in the use of Crandall's method (though hidden), his and my very fast algorithms for n^(1/n) and a modern series acceleration.

While preparing for the verifying of 7-million digits, I broke some speed records with two of my i9-14900K 6400MHZRAM (overclocked CPUs), using Mathematica 11.3 and the lightweight grid. They are generally faster than the 3 node MRB constant supercomputer with remote kernels! Unless noted by the command Timing[], i.;e., the last yellow and the final two blue-highlighted columns in the fourth table, these are all absolute timings. How does your computer compare to these? What can you do with other software?

That light blue highlight, red text columns, which shows processor times result here->7200 corrected timing for 30k and was tweaked in 2025.

For the partial column of two 6000MHz 14900K' with red text and yellow highlight, see speed 100 300 1M XMP tweaked.

For column "=F" (highlighted in green) see linked "10203050100" .

At the bottom, see attached "kernel priority 2 computers.nb" for column =B,

"3 fastest computers together.nb" for column =C

and linked "speed records 5 10 20 30 K"

also speed 50K speed 100k, speed 300k and 30p0683 hour million.nb for column =D .

For the mostly red column including the single, record, 10,114 second 300,000 digit run " =E" is in the linked "3 fastest computers together 2.nb.}

For column "=J," see 574 s 100k , .106.1 sec 50k and 6897s 300k

The 27-hour million-digit computation is found here. <-Big notebook.

As for single-device records are concerned, the fourth table: The yellow-highlighted column is documented here<- (with the others already documented above).

This is another comparison of my fastest computers' timings in calculating digits of CMRB:

The blue column (using the Wolfram Lightweight Grid) is documented here.

The i9-12900KS column is documented here.

The i9-13900KS column is documented here.

The 300,000 digits result in the i9-13900KS column is here, where it ends with the following:

  Finished on Mon 21 Nov 2022 19:55:52. Processor and actual time 
         were 6180.27 and 10114.4781964 s. respectively

  Enter MRB1 to print 301492 digits. The error from a 6,500,000 or more digit 
 calculation that used a different method is  

 Out[72]= 0.*10^-301494

Remembering that the integrated analog of the MRB constant is

NIntegrate[(-1)^n (n^(1/n) - 1), {n, 1, Infinity  I}, 
 Method -> "Trapezoidal", WorkingPrecision -> 20]

These results are from the Timing[] command:

The 14900KS at 7200 MHz (extreme tuning!) documented here

The i9-12900KS column is documented here.

"Windows10 2024 i9-14900KS 6000 MHZ RAM" documentation here

Attachments:

3 fastest comput...nb

kernel priority ...nb

Distribution of MRB constant digits as given in all record calculations

Here is the distribution of digits within the first 6,000,000 decimal places (.187859,,,), "4" shows up more than other digits, followed by "0," "8" and "7."

Here is the distribution of digits within the first 5,000,000 decimal places (.187859,,,), "4" shows up a lot more than other digits, followed by "0," "8" and "6."

Here is a similar distribution over the first 4,000,000:

3,000,000 digits share a similar distribution:

Over the first 2 and 1 million digits "4" was not so well represented. So, the heavy representation of "4" is shown to be a growing phenomenon from 2 million to 5 million digits. However, "1,2,and 5" still made a very poor showing:

I attached more than 6,000,000 digits of the MRB constant.

02/12/2019

Using my 2 nodes of the MRB constant supercomputer (3.7 GH overclocked up to 4.7 GH, Intel 6core, 3000MH RAM,and 4 cores from my 3.6 GH, 2400MH RAM) I computed 34,517 digits of the MRB constant using Crandall's first eta formula:

prec = 35000;
to = SessionTime[];
etaMM[m_, pr_] := 
  Block[{a, s, k, b, c}, 
   a[j_] := (SetPrecision[Log[j + 1], prec]/(j + 1))^m;
   {b, c, s} = {-1, -d, 0};
   Do[c = b - c;
    s = s + c a[k];
    b = (k + n) (k - n) b/((k + 1) (k + 1/2)), {k, 0, n - 1}];
   Return[N[s/d, pr] (-1)^m]];
eta1 = N[EulerGamma Log[2] - Log[2]^2/2, prec]; n = 
 Floor[132/100 prec]; d = N[ChebyshevT[n, 3], prec];
MRBtest = 
  eta1 - Total[
    ParallelCombine[((Cos[Pi #]) etaMM[#, prec]/
         N[Gamma[# + 1], prec]) &, Range[2, Floor[.250 prec]], 
     Method -> "CoarsestGrained"]];
Print[N[MRBtest2 - MRBtest,10]];

SessionTime[] - to

giving -2.166803252*10^-34517 for a difference and 208659.2864422 seconds or 2.415 days for a timing.

Where MRBtest2 is 36000 digits computed through acceleration methods of n^(1/n)

3/28/2019

Here is an updated table of speed eta formula records:

04/03/2019

Using my 2 nodes of the MRB constant supercomputer (3.7 GH overclocked up to 4.7 GH, Intel 6core, 3000MH RAM,and 4 cores from my 3.6 GH, 2400MH RAM) I computed 50,000 digits of the MRB constant using Crandall's first eta formula in 5.79 days.

 prec = 50000;
to = SessionTime[];
etaMM[m_, pr_] := 
  Module[{a, s, k, b, c}, 
   a[j_] := 
    SetPrecision[SetPrecision[Log[j + 1]/(j + 1), prec]^m, prec];
   {b, c, s} = {-1, -d, 0};
   Do[c = b - c;
    s = s + c a[k];
    b = (k + n) (k - n) b/((k + 1) (k + 1/2)), {k, 0, n - 1}];
   Return[N[s/d, pr] (-1)^m]];
eta1 = N[EulerGamma Log[2] - Log[2]^2/2, prec]; n = 
 Floor[132/100 prec]; d = N[ChebyshevT[n, 3], prec];
MRBtest = 
  eta1 - Total[
    ParallelCombine[((Cos[Pi #]) etaMM[#, prec]/
         N[Gamma[# + 1], prec]) &, Range[2, Floor[.245 prec]], 
     Method -> "CoarsestGrained"]];
Print[N[MRBtest2 - MRBtest, 10]];

SessionTime[] - to

 (* 0.*10^-50000

  500808.4835750*)

nice system!

Daniel Lichtblau and others, I just deciphered an Identity Crandall used for checking computations of the MRB constant just before he died. It is used in a previous post about checking, where I said it was hard to follow. The MRB constant is B here. B=` In input form that is

   B= Sum[(-1)^(k + 1)*(-1 + (1 + k)^(1/(1 + k)) - Log[1 + k]/(1 + k) - 
         Log[1 + k]^2/(2*(1 + k)^2)), {k, 0, Infinity}] + 
     1/24 (\[Pi]^2 Log[2]^2 - 
        2 \[Pi]^2 Log[
          2] (EulerGamma + Log[2] - 12 Log[Glaisher] + Log[\[Pi]]) - 
        6 (Zeta^\[Prime]\[Prime])[2]) + 
     1/2 (2 EulerGamma Log[2] - Log[2]^2)

For 3000 digit numeric approximation, it is

B=NSum[((-1)^(
    k + 1) (-1 + (1 + k)^(1/(1 + k)) - Log[1 + k]/(1 + k) - 
      Log[1 + k]^2/(2 (1 + k)^2))), {k, 0, Infinity}, 
  Method -> "AlternatingSigns", WorkingPrecision -> 3000] + 
 1/24 (\[Pi]^2 Log[2]^2 - 
    2 \[Pi]^2 Log[
      2] (EulerGamma + Log[2] - 12 Log[Glaisher] + Log[\[Pi]]) - 
    6 (Zeta^\[Prime]\[Prime])[2]) + 
 1/2 (2 EulerGamma Log[2] - Log[2]^2)

It is anylitaclly straight forward too because

Sum[(-1)^(k + 1)*Log[1 + k]^2/(2 (1 + k)^2), {k, 0, Infinity}]

gives

1/24 (-\[Pi]^2 (Log[2]^2 + EulerGamma Log[4] - 
      24 Log[2] Log[Glaisher] + Log[4] Log[\[Pi]]) - 
   6 (Zeta^\[Prime]\[Prime])[2])

That is I wonder why he chose it?

Crandall is not using his eta formulas directly!!!!!!! He computes Sum[(-1)^k*(k^(1/k) - 1), {k, 1, Infinity}] directly!

Going back to Crandall's code:

(*Fastest (at RC's end) as of 30 Nov 2012.*)prec = 500000;(*Number of \
required decimals.*)ClearSystemCache[];
T0 = SessionTime[];
expM[pre_] := 
  Module[{a, d, s, k, bb, c, n, end, iprec, xvals, x, pc, cores = 4, 
    tsize = 2^7, chunksize, start = 1, ll, ctab, 
    pr = Floor[1.02 pre]}, chunksize = cores*tsize;
   n = Floor[1.32 pr];
   end = Ceiling[n/chunksize];
   Print["Iterations required: ", n];
   Print["end ", end];
   Print[end*chunksize];
   d = N[(3 + Sqrt[8])^n, pr + 10];
   d = Round[1/2 (d + 1/d)];
   {b, c, s} = {SetPrecision[-1, 1.1*n], -d, 0};
   iprec = Ceiling[pr/27];
   Do[xvals = Flatten[ParallelTable[Table[ll = start + j*tsize + l;
        x = N[E^(Log[ll]/(ll)), iprec];
        pc = iprec;
        While[pc < pr, pc = Min[3 pc, pr];
         x = SetPrecision[x, pc];
         y = x^ll - ll;
         x = x (1 - 2 y/((ll + 1) y + 2 ll ll));];(*N[Exp[Log[ll]/ll],
        pr]*)x, {l, 0, tsize - 1}], {j, 0, cores - 1}, 
       Method -> "EvaluationsPerKernel" -> 1]];
    ctab = Table[c = b - c;
      ll = start + l - 2;
      b *= 2 (ll + n) (ll - n)/((ll + 1) (2 ll + 1));
      c, {l, chunksize}];
    s += ctab.(xvals - 1);
    start += chunksize;
    Print["done iter ", k*chunksize, " ", SessionTime[] - T0];, {k, 0,
      end - 1}];
   N[-s/d, pr]];

t2 = Timing[MRBtest2 = expM[prec];];
MRBtest2 - MRBtest3

x = N[E^(Log[ll]/(ll)), iprec]; Gives k^(1/k) to only Ceiling[pr/27]; decimal places; they are either 1.0, 1.1, 1.2, 1.3 or 1.4 (usually 1.1 or 1.0 for the first 27 desired decimals.) On the other hand,

While[pc < pr, pc = Min[3 pc, pr];
 x = SetPrecision[x, pc];
 y = x^ll - ll;
 x = x (1 - 2 y/((ll + 1) y + 2 ll ll));],

takes the short precision x and gives it the necessary precision and accuracy for k^(1/k) (k Is ll there.) It actually computes k^(1/k). Then he remarks, "(N[Exp[Log[ll]/ll], pr])."

After finding a fast way to compute k^(1/k) to necessary precision he uses Cohen's algorithm 1 (See a screenshot in a previous post.) to accelerate convergence of Sum[(-1)^k*(k^(1/k) - 1), {k, 1, Infinity}]. That is his secret!!

As I mentioned in a previous post the "MRBtest2 - MRBtest3" is for checking with a known-to-be accurate approximation to the MRB constant, MRBtest3

I'm just excited that I figured it out! as you can tell.

Daniel Lichtblau and others, Richard Crandall did intend to explian his work on the MRB constant and his program to compute it. When I wrote him with a possible small improvement to his program he said, "It's worth observing when we write it up." See screenshot:

What Richard Crandall and maybe others did to come up with that method is really good and somewhat mysterious. I still don't really understand the inner workings, and I had shown him how to parallelize it. So the best I can say is that it's really hard to compete against magic. (I don't want to discourage others, I'm just explaining why I myself would be reluctant to tackle this. Someone less familiar might actually have a better chance of breaking new ground.)

In a way this should be good news. Should it ever become "easy" to compute, the MRB number would lose what is perhaps its biggest point of interest. It just happens to be on that cusp of tantalizingly "close" to easily computable (perhaps as sums of zeta function and derivatives thereof), yet still hard enough that it takes a sophisticated scheme to get more than a few dozen digits.