POSTED BY: Marvin Ray Burns.

An Easter egg for you to find below:

(In another reality, I invented CMRB and then discovered many of its qualities.)

Content of the first post, as of August 16, 2022

1. My claim to the MRB constant (CMRB), or a case of calculus déjà vu?
2. What exactly is it?
3. Where is it found?
4. Q and A,
5. How it all began,
6. The why and what of the C_MRB Records,
7. C_MRB and its applications.

The following contents of the first post have been moved. Use CNRL+F to locate.

1. CNRTL+F "Real-World, and beyond, Applications".
2. CNRL+F "MeijerG Representation for" C_MRB,
3. C_MRB CNRL+F "formulas and identities",
4. CNRL+F "Primary Proof 1",
5. CNRL+F "Primary Proof 2",
6. CNRL+F "Primary Proof 3",
7. CNRL+F "The relationship between" C_MRB and its integrated analog,
8. The MRB constant supercomputer 0

Second post:

The following might help anyone serious about breaking my record.

Third post

The following email Crandall sent me before he died might be helpful for anyone checking their results.

Fourth post

Perhaps some of these speed records will be easier to beat.

Many more interesting posts

...including the MRB constant supercomputers 1 and 2.

...including records of computing the MRB constant from Crandall's eta derivative formulas.

...including all the methods used to compute C_MRB and their efficiency.

...including the dispersion of the 0-9th decimals in C_MRB decimal expansions.

...including the convergence rate of 3 major different forms of C_MRB.

...including complete documentation of all multimillion-digit records with many highlights.

...including arbitrarily close approximation formulas for C_MRB.

...including efficient programs to compute the integrated analog (MKB) of C_MRB.

...including a recent discovery that could help in verifying digital expansions of the integrated analog (MKB) of C_MRB.

...including CNRL+F "the Laplace transform analogy to" C_MRB.

...including CNRTL+F "Real-World, and beyond, Applications".

... including an overview of all C_MRB speed records, by platform.

...including 3 attempts at a spectacular 7 million digits using Mathematica.

This discussion is not crass bragging; it is an attempt by this amateur to share his discoveries with the greatest audience possible.

Amateurs have been known to make a few significant discoveries as discussed in here. This amateur has made his best attempts at proving his discoveries and has often asked for help in doing so. Great thanks to all of those who offered a hand! If I've failed to give you credit for any of your suggestions, let me know and I will correct that issue!

As I went more and more public with my discoveries, I made several attempts to see what portion of them was original. What I concluded from these investigations was that the only original thought I had was the obstinacy to think anything meaningful can be found in the infinite sum shown next. Nonetheless, it is possible that someone might have a claim to this thought to whom I have not given proper credit. If that is you I apologize. The last thing we need is another calculus war, this time for a constant. However, if your thought was published after mine, as Newton said concerning Leibniz's claim to calculus, “To take away the Right of the first inventor, and divide it between him and that other would be an Act of Injustice.” [Sir Isaac Newton, The Correspondence of Isaac Newton, 7 v., edited by H. W. Turnbull, J. F. Scott, A. Rupert Hall, and Laura Tilling, Cambridge University Press, 1959–1977: VI, p. 455]

Here is what Google says abut the MRB constant as of August 8, 2022 at https://www.google.com/search?q=who+discovered+the+%22MRB+constant%22

(the calculus war for CMRB)

CREDIT

https://soundcloud.com/cmrb/homer-simpson-vs-peter-griffin-cmrb

LATIN

PRINCIPIS CMRB MATHEMATICA

ACTA CMRB ERUDITORUM

ENGLISH

PRINCIPLES OF CMRB MATHEMATICS

REPORTS OF CMRB SCHOLARS

MATH

(Newton's notation as published in PRINCIPIS MATHEMATICA [PRINCIPLES OF MATHEMATICS])

( Leibniz's notation as published in the scholarly journal, Acta Eruditorum [Reports of Scholars])

Whether or not we divide the credit between the two pioneers,

said one thing that distinguishes their finds from the work of their antecedents:

[sic] they created a new system, like as many series have been studied for centuries leading to a labyrinth of sums, I opened up a channel for the CMRB "realities."

Newton came to calculus as part of his investigations in physics and geometry. He viewed calculus as the scientific description of the generation of motion and magnitudes. In comparison, Leibniz focused on the tangent problem and came to believe that calculus was a metaphysical explanation of the change. Importantly, the core of their insight was the formalization of the inverse properties between the integral and the differential of a function. This insight had been anticipated by their predecessors, but they were the first to conceive calculus as a system in which new rhetoric and descriptive terms were created.[24] Their unique discoveries lay not only in their imagination, but also in their ability to synthesize the insights around them into a universal algorithmic process, thereby forming a new mathematical system.

=

and from Richard Cradall in 2012 courtesy of Apple Computer's advanced computational group we have the following:

Eta denotes the kth derivative of the Dirichlet eta function of m and 0 respectively,

The c_j's are found by the code,

  N[ Table[Sum[(-1)^j Binomial[k, j] j^(k - j), {j, 1, k}], {k, 1, 10}]]

(* {-1., -1., 2., 9., 4., -95., -414., 49., 10088., 55521.}*)

The first "B" is proven below by Gottfried Helms and it is proven more rigorously, considering the conditionally convergent sum, afterward. Then the formula (44) is a Taylor expansion of eta(s) around s = 0.

At here, we have the following explanation.

Even though one has cause to be a little bit wary around formal rearrangements of conditionally convergent sums (see the Riemann series theorem), it's not very difficult to validate the formal manipulation of Helms. The idea is to cordon off a big chunk of the infinite double summation (all the terms from the second column on) that we know is absolutely convergent, which we are then free to rearrange with impunity. (Most relevantly for our purposes here, see pages 80-85 of this document. culminating with the Fubini theorem which is essentially the manipulation Helms is using.)

So, by definition the MRB constant $B$ is the conditionally convergent sum $\sum_{n \geq 1} (-1)^n (n^{1/n} - 1)$. Put $a_n = (-1)^n (n^{1/n}= - 1)$,

so $B = \sum_{n \geq 1} a_n.$

Looking at the first column, put $b_n = (-1)^n \frac{\log(n)}{n}.$

so $\eta^{(1)}(1) = \sum_{n \geq 1}> b_n$

as a conditionally convergent series.

We have

$$B - \eta^{(1)} = \sum_{n \geq 1} a_n - b_n = \sum_{n \geq 1} \sum_{m \geq 2} (-1)^n \frac{(\log n)^m}{n^m m!}$$

(The first equation is an elementary limit statement that says if $\sum_{n \geq 1} a_n$ converges and $\sum_{n \geq 1} b_n$ converges, then also $\sum_{n \geq 1} a_n - b_n$ converges and $\sum_{n \geq 1} a_n - \sum_{n \geq 1} b_n = \sum_{n \geq 1} a_n - b_n$. It doesn't at all matter whether the convergence of either series is conditional or absolute.)

So now we check the absolute convergence of the right-hand side, i.e., that $\sum_{n \geq 1} \sum_{m \geq 2} \frac{(\log n)^m}{n^m m!}$ converges. (Remember what this means in the case of infinite sums of positive terms: it means that there is a number $K$ such that every finite partial sum $S$ is bounded above by $K$; the least such upper bound will be the number that the infinite sum converges to.) So take any such finite partial sum $S$, and rearrange its terms so that the terms in the $m = 2$ column come first, then the terms in the $m = 3$ column, and so on. An upper bound for the terms of $S$ in the $m = 2$ column is $\frac{\zeta^{(2)}(2)}{2!}$. Put that one aside.

For the $m = 3$ column, an upper bound is $\sum_{n \geq 2} \frac{(\log n)^3}{n^3 3!}$ (we drop the $n=1$ term which is $0$). By calculus we have $\log n \leq n^{1/2}$ for all $n \geq 2$, so this has upper bound $\frac1{3!} \sum_{n \geq 2} \frac1{n^{3/2}} \leq \frac1{3!} \int_1^\infty \frac{dx}{x^{3/2}}$ by an integral test, which yields $\frac{2}{3!}$ as an upper bound. Applying the same reasoning for the $m$ column from $m = 4$ on, an upper bound for that column would be $\frac1{m!} \int_1^\infty \frac{dx}{x^{m/2}} = \frac{2}{m!(m-2)}$. Adding all those upper bounds together, an upper bound for the entire doubly infinite sum would be

$$\frac{\zeta^{(2)}(2)}{2!} + \sum_{m \geq 3} \frac{2}{m!(m-2)}$$

which certainly converges. So we have absolute convergence of the doubly infinite sum.

Thus we are in a position to apply the Fubini theorem, which justifies the rearrangement expressed in the first of the following equations

$$\sum_{n \geq 1} \sum_{m \geq 2} (-1)^n \frac{(\log n)^m}{n^m m!} = \sum_{m \geq 2} \sum_{n \geq 1} (-1)^n \frac{(\log n)^m}{n^m m!} = \sum_{m \geq 2} (-1)^{m+1} \frac{\eta^{(m)}(m)}{m!}$$

giving us what we wanted.

​​

​​​​​​

​​

Examples

In[]:=

Clear[t];ChopCMRB-N

10^10

∑

n=1



1

n

n

Cos[π(n+2)],​​

∞

∑

t=1

-1+

1

t

t

Sec[π(t+2)],​​ Im

InfinityI

∫

1

-1+

1

t

t

Sec[π(1/2+t+2)]t,10^-8//Quiet

Out[]=

{0,0,0}

CMRB=N[NSum[(-1)^n(n^(1/n)-1),{n,1,Infinity},Method->"AlternatingSigns",WorkingPrecision->37],30];Tableα:=a-x;β:=b-x;​​t:=1+

β

α

;f[x_]=

1

α

;g[x_]=

f[x]

f[b]

;n:=f[x]β;​​CMRB-AbsNIntegrate((Re[t^(1/t)*(Csc[Pi*(n+2)])]))

2

f[x]

f[b]

,{x,a,b},WorkingPrecision->30,​​​​Style[CMRB-Abs[NIntegrate[((Re[t^(1/t)*(Sec[Pi*(t+2-1/2)])]))g'[x],{x,a,b},WorkingPrecision->30]],FontColor->Blue]​​​​​​​​//Chop,{a,-2,2},{b,a+1,3}//TableForm

Out[]//TableForm=

0 0	0 0	0 0	0 0	0 0
0 0	0 0	0 0	0 0
0 0	0 0	0 0
0 0	0 0
0 0

is defined in all of the following places, most of which attribute it to my curiosity:

• ค่าคงที่ลุ่มแม่น้ำโขง (in Thai);

• ar.wikipedia.org/wiki/ (In Arabic);

• Constante MRB (in French);

• Constanta MRB - MRB constant (in Romanian);

• http://constant.one/ ;

• Crandall, R. E. "The MRB Constant." §7.5 in Algorithmic Reflections: Selected Works. PSI Press, pp. 28-29, 2012,ISBN-10 : ‎193563819X ISBN-13: ‎978-1935638193;

• Crandall, R. E. "Unified Algorithms for Polylogarithm, L-Series, and Zeta Variants." 2012;

• https://en-academic.com/, Wikipedia, Mathematical constant;

• Encyclopedia of Mathematics (Series #94);

• Engineering Tools of the Iran Civil Center, (translated from Persian), an international community dedicated to the construction industry, ISSN: 1735–2614;

• Etymologie CA Kanada Zahlen" (in German). etymologie.info;

• Finch, S. R. Mathematical Constants, Cambridge, England: Cambridge University Press, p. 450, 2003, ISBN-13: 978-0521818056, ISBN-10: 0521818052;

• Finch's original essay on Iterated Exponential Constants;

• Finch, Steven & Wimp, Jet. (2004). Mathematical constants. The Mathematical Intelligencer. 26. 70-74. 10.1007/BF02985660;

• Journal of Mathematics Research; Vol. 11, No. 6; December 2019 ISSN 1916-9795 E-ISSN 1916-9809 Published by Canadian Center of Science and Education;

• Mauro Fiorentina’s math notes (in Italian);

• MATHAR, RICHARD J. "NUMERICAL EVALUATION OF THE OSCILLATORY INTEGRAL OVER exp(iπx) x^(1/x) BETWEEN 1 AND INFINITY" (PDF). arxiv. Cornell University;

• Mathematical Constants and Sequences a selection compiled by Stanislav Sýkora, Extra Byte, Castano Primo, Italy. Stan’s Library, ISSN 2421-1230, Vol.II;

• "Matematıksel Sabıtler" (in Turkish). Türk Biyofizik Derneği;

• MathWorld Encyclopedia;

• MRB常数 (in Chinese);

• mrb constantとは 意味・読み方・使い方 ( in Japanese);

• MRB константа (in Bulgarian);

• OEIS Encyclopedia (The MRB constant);

• Patuloy ang MRB - MRB constant (in Filipino)

• Plouffe's Inverter;

• the LACM Inverse Symbolic Calculator;

• The On-Line Encyclopedia of Integer Sequences® (OEIS®) as A037077, Notices Am. Math. Soc. 50 (2003), no. 8, 912–915, MR 1992789 (2004f:11151);

• Wikipedia Encyclopedia.

Q&A

Q: What can you expect from reading about C_MRB and its record computations?

A:

As you see, the war treated me kindly enough, in spite of the heavy gunfire, to allow me to get away from it all and take this walk in the land of your ideas.

— Karl Schwarzschild (1915), “Letter to Einstein”, Dec 22

Q: Can you calculate more digits of the MRB constant?

A:

With the availability of high-speed electronic computers, it is now quite convenient to devise statistical experiments for the purpose of estimating certain mathematical constants and functions.

Copyright © 1966 ACM
(Association for Computing Machinery)

New York, NY, United States

Q: How can you compute them?

A:

The value of $\pi$ has engaged the attention of many mathematicians and calculators from the time of Archimedes to the present day, and has been computed from so many different formulae, that a complete account of its calculation would almost amount to a history of mathematics.

- James Glaisher (1848-1928)

Q: Why should you do it?

A:

While it is never safe to affirm that the future of Physical Science has no marvels in store even more astonishing than those of the past, it seems probable that most of the grand underlying principles have been firmly established and that further advances are to be sought chiefly in the rigorous application of these principles to all the phenomena which come under our notice. It is here that the science of measurement shows its importance — where quantitative work is more to be desired than qualitative work. An eminent physicist remarked that the future truths of physical science are to be looked for in the sixth place of decimals.

Albert A. Michelson (1894)

Q: Why are those digits there?

A:

There is nothing without a reason.

Read more at: https://minimalistquotes.com/gottfried-leibniz-quote-229585/

How it all began

From these meager beginnings: My life has proven that one's grades in school are not necessarily a prognostication of achievement in mathematics. For evidence of my poor grades see my report cards.

The eldest child, raised by my sixth-grade-educated mother, I was a D and F student through 6th grade the second time, but in Jr high, in 1976, we were given a self-paced program. Then I noticed there was more to math than rote multiplication and division of 3 and 4-digit numbers! Instead of repetition, I was able to explore what was out there. The more I researched, the better my grades got! It was amazing!! So, having become proficient in mathematics during my high school years, on my birthday in 1994, I decided to put down the TV remote control and pick up a pencil. I began by writing out the powers of 2, like 2*2, 2*2*2, etc. I started making up algebra problems to work at solving, and even started buying books on introductory calculus.
Then came my first opportunity to attend university. I took care of my mother, who suffered from Alzheimer's, so instead of working my usual 60+ hours a week. I started taking a class or two a semester. After my mom passed away, I went back to working my long number of hours but always kept up on my math hobby! 
Occasionally, I make a point of going to school and taking a class or two to enrich myself and my math hobby. This has become such a successful routine that some strangers listed me on Wikipedia as an amateur mathematician alphabetically following Jost Bürgi who constructed a table of progressions that is now understood as antilogarithms independently of John Napier at the behest of Johannes Kepler.
I've even studied a few graduate-level topics in Mathematics.

Why I started so slow and am now a pioneer is a mystery! Could it say something about the educational system? Can the reason be found deep in psychology? (After all, I never made any progress in math or discoveries without first assuming I could, even when others were telling me, I couldn't!) Or could it be that the truth is a little of both and more?

From these meager beginnings:

On January 11 and 23,1999 I wrote,

I have started a search for a new mathematical constant! Does anyone want to help me? Consider, 1^(1/1)-2^(1/2)+3^(1/3)...I will take it apart and examine it "bit by bit." I hope to find connections to all kinds of arithmetical manipulations. I realize I am in "no man's land," but I work best there! If anyone else is foolhardy enough to come along and offer advice, I welcome you.

The point is that I found the MRB constant (C_MRB), meaning after all the giants were through roaming the forest of numbers and founding all they found, one virgin mustard seedling caught my eye. So I carefully "brought it up" to a level of maturity, and my own understanding of math along with it! (In another reality, I invented C_MRB and then discovered many of its qualities.)

In doing so, I came to find out that this constant (C_MRB)

(from https://mathworld.wolfram.com/MRBConstant.html)

was more closely related to other constants than I could have imagined.

As the apprentice of all, building upon the foundation of Chebyshev (1854–1859) on the best uniform approximation of functions, as vowed on January 23, 1999. "I took C_MRB apart and examined it 'bit by bit,' finding connections to all kinds of arithmetical manipulations." Not satisfied with conveniently construed constructions (half-hazardously put together formulas) that a naïve use of numeric search engines like Wolfram Alpha or the OEIS, might give, I set out to determine the most interesting (by being the most improbable but true) approximations for each constant in relation to it.

For example, consider its relationship to Viswanath's constant (VC)

(from https://mathworld.wolfram.com/RandomFibonacciSequence.html)

With both being functions of x^1/x alone, we have these near-zeors of VC using C_MRB, which have a ratio of Gelfond's constant $=e^\pi.$ Notice, by "near-zeros," I mean we have the following.

VC/(6*(11/7 - ProductLog[1])) - CMRB
CMRB - (5*VC^6)/56

3.4164*10^-8

1.47*10^-9

Out[54]= 3.4164*^-8

Out[55]= 1.47*^-9

See cloud notebook.

Then there is the Rogers - Ramanujan Continued Fraction, R(q),

of C_MRB that is well-linearly-approximated by terms of itself alone:

From these meager beginnings:

On Feb 22, 2009, I wrote,

It appears that the absolute value, minus 1/2, of the limit(integral of (-1)^x*x^(1/x) from 1 to 2N as N->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? 

I came to find out my discovery, a very slow to converge oscillatory integral, would later be further defined by Google Scholar.

Here is proof of a faster converging integral for its integrated analog (The MKB constant) by Ariel Gershon.

g(x)=x^(1/x), M1=

Which is the same as

because changing the upper limit to 2N + 1 increases MI by $2i/\pi.$

MKB constant calculations have been moved to their discussion at http://community.wolfram.com/groups/-/m/t/1323951?ppauth=W3TxvEwH .

Plugging in equations [5] and [6] into equation [2] gives us:

Now take the limit as N?? and apply equations [3] and [4] : He went on to note that

After I mentioned it to him, Richard Mathar published his meaningful work on it here in arxiv, where M is the MRB constant and M1 is MKB:

M1 has a convergent series, which has lines of symmetry across whole-and-half number points on the x-axis, and half-periods of exactly 1, for both real and imaginary parts as in the following plots. And where

f[x_] = Exp[I Pi x] (x^(1/x) - 1); Assuming[
 x \[Element] Integers && x > 1, 
 FullSimplify[Re[f[x + 1/2]] - Im[f[x]]]]

gives 0

ReImPlot[(-1)^x (x^(1/x) - 1), {x, 1, Infinity}, PlotStyle -> Blue, 
 Filling -> Axis, FillingStyle -> {Green, Red}]

Every 75 i of the upper value of the partial integration yields 100 additional digits of M2= and of CMRB==

Here is a heuristic explanation for the observed behavior.

Write the integral as an infinite series, $m= \sum_{k = 1}^\infty a_k$ with $a_k = \int_{i kM}^{i (k+1)M} \frac{t^{1/t}-1}{\sin (\pi t)} \, dt$ for $k \ge 2$ and the obvious modification for $k = 1$. we are computing the partial sums of these series with $M = 75$ and the question is why the series remainders decrease by a factor of $10^{-100}$ for each additional term.

The integrand is a quotient with numerator $t^{1/t} - 1 \approx \log t\, / t$ and denominator $1/\sin \pi t \approx e^{i \pi t}$ for large imaginary $t$. The absolute values of these terms therefore are $|a_k| \approx \log |kM|/|kM| \cdot e^{-\pi kM}$. This implies

as $k \to \infty$. Consequently the remainders $\sum_{k = N}^\infty$ behave like $e^{- \pi N M}$. They decrease by a factor of $e^{-\pi M}$ for each additional term. And for $M = 75$, this is approximately $10^{-100}$, predicting an increase in accuracy of 100 digits whenever the upper integration bound increased by $75i$.

I used the fact that

The following "partial proof of it" is from Quora.

While

I developed a lot more theory behind it and ways of computing many more digits in this linked Wolfram post.

Here is how my analysis (along with improvements to Mathematica) has improved the speed of calculating that constant's digits:

(digits and seconds)

Better 2022 results are expected soon!

2022 results documentation:

In[]:=

Timing[​​MKB-Quiet[NIntegrate[​​Exp[PiIx]Sum[(Log[x]/x)^n/n!,{n,1,Infinity}],{x,1,​​InfinityI},WorkingPrecision1000,Method"Trapezoidal"]]]

Out[]=

{1.8125,0.×

-1002

10

+0.×

-1002

10

}

In[]:=

Timing[​​MKB-Quiet[NIntegrate[​​Exp[PiIx]Sum[(Log[x]/x)^n/n!,{n,1,Infinity}],{x,1,​​InfinityI},WorkingPrecision2000,Method"Trapezoidal",​​MaxRecursion10]]]

Out[]=

{12.4844,0.×

-2002

10

+0.×

-2002

10

}

In[]:=

Timing[​​MKB-Quiet[NIntegrate[​​Exp[PiIx]Sum[(Log[x]/x)^n/n!,{n,1,Infinity}],{x,1,​​InfinityI},WorkingPrecision3000,Method"Trapezoidal",​​MaxRecursion11]]]

Out[]=

{53.4063,0.×

-3002

10

+0.×

-3002

10

}

In[]:=

Timing[​​MKB-Quiet[NIntegrate[​​Exp[PiIx]Sum[(Log[x]/x)^n/n!,{n,1,Infinity}],{x,1,​​InfinityI},WorkingPrecision4000,Method"Trapezoidal",​​MaxRecursion11]]]

Out[]=

{94.375,0.×

-4002

10

+0.×

-4002

10

}

In[]:=

Timing[​​MKB-Quiet[NIntegrate[​​Exp[PiIx]Sum[(Log[x]/x)^n/n!,{n,1,Infinity}],{x,1,​​InfinityI},WorkingPrecision5000,Method"Trapezoidal",​​MaxRecursion12]]]

Out[]=

{285.813,0.×

-5002

10

+0.×

-5002

10

}

In[]:=

ClearSystemCache[];Timing[​​MKB-Quiet[NIntegrate[​​Exp[PiIx]Sum[(Log[x]/x)^n/n!,{n,1,Infinity}],{x,1,​​InfinityI},WorkingPrecision10000,Method"Trapezoidal",​​MaxRecursion13]]]

Out[]=

{2204.64,0.×

-10002

10

+0.×

-10002

10

}

ClearSystemCache[];Timing[​​MKB-Quiet[NIntegrate[​​Exp[PiIx]Sum[(Log[x]/x)^n/n!,{n,1,Infinity}],{x,1,​​InfinityI},WorkingPrecision40000,Method"Trapezoidal",​​MaxRecursion15]]]

Out[]=

{114362.,0.×

-40002

10

+0.×

-40002

10

}

In[]:=

ClearSystemCache[];Timing[​​MKB-Quiet[NIntegrate[​​Exp[PiIx]Sum[(Log[x]/x)^n/n!,{n,1,Infinity}],{x,1,​​InfinityI},WorkingPrecision1000,Method"Trapezoidal"]]]

Out[]=

{2.04688,0.×

-1002

10

+0.×

-1002

10

}

From these meager beginnings:

In October 2016, I wrote the following here in researchgate:

First, we will follow the path the author took to find out that for

the limit of the ratio of a to a - 1, as a goes to infinity is Gelfond's Constant, (e ^ pi). We will consider the hypothesis and provide hints for proof using L’ Hospital’s Rule (since we have indeterminate forms as a goes to infinity):

The following should help in a proof of the hypothesis:

Cos[PiIx] == Cosh[Pix], Sin[PiIx] == I Sin-h[Pix], and Limit[x^(1/x),x->Infinity]==1.

Using L’Hospital’s Rule, we have the following:

(17) (PDF) Gelfond' s Constant using MKB constant like integrals. Available from: https://www.researchgate.net/publication/309187705Gelfond%27sConstantusingMKBconstantlikeintegrals [accessed Aug 16 2022].

We find there is no limit a goes to infinity, of the ratio of the previous forms of integrals when the "I" is left out, and give a small proof for their divergence.

That was responsible for the integral-equation-discovery mentioned in one of the following posts, where it is written, "Using those ratios, it looks like" (There is the MRB constant.)

From these meager beginnings:

In November 2013, I wrote:

$C$MRB is approximately 0.1878596424620671202485179340542732. See this and this.

$\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-a)$ is formally convergent only when $a =1$. However, if you extend the meaning of $\sum$ through "summation methods", whereby series that diverge in one sense converge in another sense (e.g. Cesaro, etc.) you get results for other $a$. A few years ago it came to me to ask what value of $a$ gives $$\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-a)=0\text{ ?}$$(For what value of a is the Levin's u-transform's and Cesàro's sum result 0 considering weak convergence?)

The solution I got surprised me: it was $a=1-2\times C\mathrm{MRB}=0.6242807150758657595029641318914535398881938101997224\ldots$. Where $C\mathrm{MRB}$ is $\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-1)$.

I asked, "If that's correct can you explain why?" and got the following comment.

To see this for yourself in Mathematica enter FindRoot[NSum[(-1)^n*(n^(1/n) - x), {n, 1, Infinity}], {x, 1}] where regularization is used so that the sum that formally diverges returns a result that can be interpreted as evaluation of the analytic extension of the series.

Finally let a = M2 = $1-2\times C$MRB = 0.6242807150758... and the two limit-points of the series $\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-M2)$ are +/- $C$MRB with its Levin's u-transform's result being 0. See here.

Also,

Scholarly works about C_MRB.

From these meager beginnings:

In 2015 I wrote:

Mathematica makes some attempts to project hyper-dimensions onto 2-space with the Hypercube command. Likewise, some attempts at tying them to our universe are mentioned at https://bctp.berkeley.edu/extraD.html . The MRB constant from infinite-dimensional space is described at http://marvinrayburns.com/ThegeometryV12.pdf. It is my theory that like the MRB constant the universe, under inflation, started in an infinite number of space dimensions. And they almost all instantly collapsed, as far as our sensory realm is concerned, leaving all but the few we enjoy today.

I'm not the first person to think the universe consists of an infinitude of dimensions. Some string theories and theorists propose it too. Michele Nardelli added the following.

In string theory, perturbation methods involve such a high degree of approximation that the theory is unable to identify which of the Calabi - Yau spaces are candidates for describing the universe. The consequence is that it does not describe a single universe, but something like 10^500 universes. In reality, admitting 10^500 different quantum voids would allow the only mechanism known at the moment to explain the present value of the cosmological constant following an idea by Steven Weinberg. Furthermore, a very large value of different voids is typical of any type of matter coupled to gravity and is also obtained when coupling the standard model. I believe that the multiverse is a "space of infinite dimensions" with infinite degrees of freedom and infinite possible potential wave functions that when they collapse, formalize a particle or a universe in a quantum state. The strings vibrating, like the strings of a musical instrument, emit frequencies that are not always precise numbers, indeed, very often they are decimal, irrational, and/or transcendent numbers. The MRB constant serves as a "regularizer" to obtain solutions as precise as possible and this in various sectors of string theory, black holes, and cosmology

In this physics.StackExchange question his concept of the dimensions in string theory and a possible link with number theory is inquired about.

Many MRB constant papers by Michele Nardelli are found here in Google Scholar, which include previous versions of these.

Hello. Here are the links of my more comprehensive articles describing the various applications of the CMRB in various fields of theoretical physics and cosmology. Thanks always for your availability, see you soon.

--

Analyzing several equations concerning Geometric Measure Theory, various Ramanujan parameters and the developments of the MRB Constant. New possible mathematical connections with some sectors of String Theory. XII

--

On several equations concerning Geometric Measure Theory, various Ramanujan parameters and the developments of the MRB Constant. New possible mathematical connections with some sectors of Cosmology (Bubble universes) and String Theory

--

Analyzing several equations concerning Geometric Measure Theory, various Ramanujan parameters and the developments of the MRB Constant. New possible mathematical connections with some sectors of Cosmology (Bubble universes) and String Theory. III

--

Analyzing several equations concerning Geometric Measure Theory, various Ramanujan parameters and the developments of the MRB Constant. New possible mathematical connections with some equations concerning various aspects of Quantum Mechanics and String Theory. VI

--

Analyzing several equations concerning various aspects of String Theory and one-loop graviton correction to the conformal scalar mode function. New possible mathematical connections with various Ramanujan parameters and some developments of the MRB Constant.

--

Analyzing several equations concerning Geometric Measure Theory, various Ramanujan parameters and the developments of the MRB Constant. New possible mathematical connections with some equations concerning Multiverse models and the Lorentzian path integral for the vacuum decay process

--

On the study of various equations concerning Primordial Gravitational Waves in Standard Cosmology and some sectors of String Theory. New possible mathematical connections with various Ramanujan formulas and various developments of the MRB Constant

--

On the study of some equations concerning the mathematics of String Theory. New possible connections with some sectors of Number Theory and MRB Constant

--

On the study of some equations concerning the mathematics of String Theory. New possible connections with some sectors of Number Theory and MRB Constant. II

--

Analyzing some equations of Manuscript Book 2 of Srinivasa Ramanujan. New possible mathematical connections with several equations concerning the Geometric Measure Theory, the MRB Constant and various sectors of String Theory

--

Analyzing the MRB Constant in Geometric Measure Theory and in a Ramanujan equation. New possible mathematical connections with ζ(2), ϕ , the Quantum Cosmological Constant and some sectors of String Theory

--

Analyzing some equations of Manuscript Book 2 of Srinivasa Ramanujan. New possible mathematical connections with several equations concerning the Geometric Measure Theory, the MRB Constant, various sectors of Black Hole Physics and String Theory

--

Analyzing further equations of Manuscript Book 2 of Srinivasa Ramanujan. New possible mathematical connections with the MRB Constant, the Ramanujan-Nardelli Mock General Formula and several equations concerning some sectors of String Theory III

His latest papers on the MRB constant, follow.

Hi Marvin, for me the best links you could post are those related to the works concerning the Ramanujan continued fractions and mathematical connections with MRB Constant and various sectors of String Theory.

Here are the links (in all there are 40):

https://www.academia.edu/80247977/ https://www.academia.edu/80298701/ https://www.academia.edu/80376615/ https://www.academia.edu/80431963/ https://www.academia.edu/80508286/ https://www.academia.edu/80590932/ https://www.academia.edu/80660709/ https://www.academia.edu/80724379/ https://www.academia.edu/80799006/ https://www.academia.edu/80894850/ https://www.academia.edu/81033980/ https://www.academia.edu/81150262/ https://www.academia.edu/81231887/ https://www.academia.edu/81313294/ https://www.academia.edu/81536589/ https://www.academia.edu/81625054/ https://www.academia.edu/81705896/ https://www.academia.edu/81769347/ https://www.academia.edu/81812404/ https://www.academia.edu/81874954/ https://www.academia.edu/81959191/ https://www.academia.edu/82036273/ https://www.academia.edu/82080277/ https://www.academia.edu/82129372/ https://www.academia.edu/82155422/ https://www.academia.edu/82204999/ https://www.academia.edu/82231273/ https://www.academia.edu/82243774/ https://www.academia.edu/82347058/ https://www.academia.edu/82399680/ https://www.academia.edu/82441768/ https://www.academia.edu/82475969/ https://www.academia.edu/82516896/ https://www.academia.edu/82521506/ https://www.academia.edu/82532215/ https://www.academia.edu/82622577/ https://www.academia.edu/82679726/ https://www.academia.edu/82733681/ https://www.academia.edu/82777895/ https://www.academia.edu/82828901/

He recently added the following.

Hi Marvin,

The MRB Constant, also in the case of the Ramanujan's expressions that we are slowly analyzing, serves to "normalize", therefore to rectify the approximations we obtain. For example, for the value of zeta (2), which is always approximate (1.64382 ....)" [from the string theory equations, example below], "adding an expression containing the MRB Constant gives a result much closer to the real value which is 1.644934 ... This procedure is carried out on all those we call "recurring numbers" (Pi, zeta (2), 4096, 1729 and the golden ratio), which, developing the expressions, are always approximations, from which, by inserting the CMRB in various ways, we obtain results much closer to the real values of the aforementioned recurring numbers. Finally, remember that Ramanujan's expressions and the recurring numbers that are obtained are connected to the frequencies of the strings, therefore to the vibrations of the same.

One example of his procedure from https://www.academia.edu/81812404/OnfurtherRamanujanscontinuedfractionsmathematicalconnectionswithMRBConstantvariousequationsconcerningsomesectorsofStringTheoryXIX?

was to analyze some expressions from Ramanujan's notebooks.

Finding other expressions from series of their anti-derivative and derivatives, in this case, dividing two previous expressions, after some calculations, he obtained this expression from it."

Then finally "by inserting the CMRB, obtaining results much closer to the real values of the aforementioned recurring numbers:" (referring to Ramanujan's equation, and the after more work..)

,

You need to look at the paper entirely to see how he puts it all together. He uses Wolfram Alpha for a lot of it.

7/7/2022 I just found a video he made concerning his work on string theory with its connection to Ramanujan and C_MRB. English subtitles are available on youtube.

More Google Scholar results on C_MRBt are here, which include the following.

Dr. Richard Crandall called the MRB constant a key fundamental constant

in this linked well-sourced and equally greatly cited Google Scholar promoted paper. Also here.

Dr. Richard J. Mathar wrote on the MRB constant here.

Xun Zhou, School of Water Resources and Environment, China University of Geosciences (Beijing), wrote the following in "on Some Series and Mathematic Constants Arising in Radioactive Decay" for the Journal of Mathematics Research, 2019.

A divergent infinite series may also lead to mathematical constants if its partial sum is bounded. The Marvin Ray Burns’ (MRB) constant is the upper bounded value of the partial sum of the divergent and alternating infinite series: -1^1/1+2^1/2-3^1/3+4^1/4-5^1/5+6^1/6-···=0.187859···(M. Chen, & S. Chen, 2016). Thus, construction of new infinite series has the possibility of leading to new mathematical constants.

MRB Constant Records,

My inspiration to compute a lot of digits of C_MRB came from the following website by Simon Plouffe.

There, computer mathematicians calculate millions, then billions of digits of constants like pi, when with only 65 decimal places of pi, we could determine the size of the observable universe to within a Planck length (where the uncertainty of our measure of the universe would be greater than the universe itself)!

In contrast 65 digits of the MRB constant "measures" the value of -1+ssqrt(2)-3^(1/3) up to n^(1/n) where n is 1,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000,000, which can be called 1 unvigintillion or just 10^66.

And why compute 65 digits of the MRB constant? Because having that much precision is the only way to solve such a problem as 

1465528573348167959709563453947173222018952610559967812891154^ m-m, where m is the MRB constant, which gives the near integer "to beat all," 200799291330.9999999999999999999999999999999999999999999999999999999999999900450...

And why compute millions of digits of it? uhhhhhhhhhh.... "Because it's there!" (...Yeah, thanks George Mallory!)
And why?? (c'est ma raison d'être!!!)

So, below you find reproducible results with methods. The utmost care has been taken to assure the accuracy of the record number of digits calculations. These records represent the advancement of consumer-level computers, 21st century Iterative methods, and clever programming over the past 23 years.

Here are some record computations of C_MRB. If you know of any others let me know, and I will probably add them!

1 digit of the (additive inverse of ) **C**<sub>*MRB*</sub> with my TI-92s, by adding 1-sqrt(2)+3^(1/3)-4^(1/4)+... as far as I could, was computed. That first digit, by the way, was just 0. Then by using the sum key, to compute $\sum _{n=1}^{1000 } (-1)^n \left(n^{1/n}\right),$  the first correct decimal of $\text{CMRB}=\sum _{n=1}^{\infty } (-1)^n \left(n^{1/n}-1\right)$ i.e. (.1). It gave (.1_91323989714) which is close to what Mathematica gives for summing to only an upper limit of 1000. 

4 decimals(.1878) of CMRB were computed on Jan 11, 1999 with the Inverse Symbolic Calculator, applying the command evalf( 0.1879019633921476926565342538468+sum((-1)^n* (n^(1/n)-1),n=140001..150000)); where 0.1879019633921476926565342538468 was the running total of t=sum((-1)^n* (n^(1/n)-1),n=1..10000), then t= t+the sum from (10001.. 20000), then t=t+the sum from (20001..30000) ... up to t=t+the sum from (130001..140000).

5 correct decimals (rounded to .18786), in Jan of 1999 were drawn from CMRB using Mathcad 3.1 on a 50 MHz 80486 IBM 486 personal computer operating on Windows 95.

 9 digits of CMRB shortly afterward using Mathcad 7 professional on the Pentium II mentioned below, by summing (-1)^x x^(1/x) for x=1 to 10,000,000, 20,000,000, and a many more, then linearly approximating the sum to a what a few billion terms would have given.

 500 digits of CMRB with an online tool called Sigma on Jan 23, 1999. See [http://marvinrayburns.com/Original_MRB_Post.html][130]   if you can read the printed and scanned copy there.

5,000 digits of CMRB in September of 1999 in 2 hours on a 350 MHz PentiumII,133 MHz 64 MB of RAM using the simple PARI commands \p 5000;sumalt(n=1,((-1)^n*(n^(1/n)-1))), after allocating enough memory.
To beat that, I did it on July 4, 2022, in 1 second on the 5.5 GHz CMRBSC 3 with 4800MHz 64 GB of RAM by Newton's method using Convergence acceleration of alternating series. Henri Cohen, Fernando Rodriguez Villegas, Don Zagier acceleration "Algorithm 1" to at least 5000 decimals. (* Newer loop with Newton interior. *)

documentation here

 6,995 accurate digits of CMRB were computed on June 10-11, 2003 over a period, of 10 hours, on a 450 MHz P3 with an available 512 MB RAM,.
To beat that, I did it in <2.5 seconds on the MRBCSC 3 on July 7, 2022 (more than 14,400 times as fast!)

documentation here

8000 digits of CMRB completed, using a Sony Vaio P4 2.66 GHz laptop computer with 960 MB of available RAM, at 2:04 PM 3/25/2004,

  11,000 digits of CMRB> on March 01, 2006, with a 3 GHz PD with 2 GB RAM available calculated.

 40, 000 digits of CMRB in 33 hours and 26 min via my program written in Mathematica 5.2 on Nov 24, 2006. The computation was run on a 32-bit Windows 3 GHz PD desktop computer using 3.25 GB of Ram.
The program was

    Block[{a, b = -1, c = -1 - d, d = (3 + Sqrt[8])^n, 
      n = 131 Ceiling[40000/100], s = 0}, a[0] = 1;
     d = (d + 1/d)/2; For[m = 1, m < n, a[m] = (1 + m)^(1/(1 + m)); m++];
     For[k = 0, k < n, c = b - c;
      b = b (k + n) (k - n)/((k + 1/2) (k + 1)); s = s + c*a[k]; k++];
     N[1/2 - s/d, 40000]]

 60,000 digits of CMRB on July 29, 2007, at 11:57 PM EST in 50.51 hours on a 2.6 GHz AMD Athlon with 64-bit Windows XP. Max memory used was 4.0 GB of RAM.

65,000 digits of CMRB in only 50.50 hours on a 2.66 GHz Core 2 Duo using 64-bit Windows XP on Aug 3, 2007, at 12:40 AM EST, Max memory used was 5.0 GB of RAM.

100,000 digits of CMRB on Aug 12, 2007, at 8:00 PM EST, were computed in 170 hours on a 2.66 GHz Core 2 Duo using 64-bit Windows XP. Max memory used was 11.3 GB of RAM. The typical daily record of memory used was 8.5 GB of RAM.
To beat that, on the 4th of July, 2022, I computed the same digits in  1/4 of an hour. CNTRL+F "4th of July, 2022" for documentation.
To beat that, on the 7th of July, 2022, I computed the same digits in  1/5 of an hour. CNTRL+F "7th of July, 2022" for documentation (850 times as fast as the first 100,000 run!)

 150,000 digits of CMRB on Sep 23, 2007, at 11:00 AM EST. Computed in 330 hours on a 2.66 GHz Core 2 Duo using 64-bit Windows XP. Max memory used was 22 GB of RAM. The typical daily record of memory used was 17 GB of RAM.

  200,000 digits of CMRB using Mathematica 5.2 on March 16, 2008, at 3:00 PM EST,. Found in 845 hours, on a 2.66 GHz Core 2 Duo using 64-bit Windows XP. Max memory used was 47 GB of RAM. The typical daily record of memory used was 28 GB of RAM.

300,000 digits of CMRB were destroyed (washed away by Hurricane Ike ) on September 13, 2008 sometime between 2:00 PM - 8:00 PM EST. Computed for a long  4015. Hours (23.899 weeks or 1.4454*10^7 seconds) on a 2.66 GHz Core 2 Duo using 64-bit Windows XP. Max memory used was 91 GB of RAM. The Mathematica 6.0 code is used follows:

    Block[{$MaxExtraPrecision = 300000 + 8, a, b = -1, c = -1 - d, 
     d = (3 + Sqrt[8])^n, n = 131 Ceiling[300000/100], s = 0}, a[0] = 1; 
     d = (d + 1/d)/2; For[m = 1, m < n, a[m] = (1 + m)^(1/(1 + m)); m++]; 
     For[k = 0, k < n, c = b - c; 
      b = b (k + n) (k - n)/((k + 1/2) (k + 1)); s = s + c*a[k]; k++]; 
     N[1/2 - s/d, 300000]]

225,000 digits of CMRB were started with a 2.66 GHz Core 2 Duo using 64-bit Windows XP on September 18, 2008. It was completed in 1072 hours. 

250,000 digits were attempted but failed to be completed to a serious internal error that restarted the machine. The error occurred sometime on December 24, 2008, between 9:00 AM and 9:00 PM. The computation began on November 16, 2008, at 10:03 PM EST. The Max memory used was 60.5 GB.

 250,000 digits of CMRB on Jan 29, 2009, 1:26:19 pm (UTC-0500) EST, with a multiple-step Mathematica command running on a dedicated 64-bit XP using 4 GB DDR2 RAM onboard and 36 GB virtual. The computation took only 333.102 hours. The digits are at http://marvinrayburns.com/250KMRB.txt. The computation is completely documented.

  300000 digit search of CMRB was initiated using an i7 with 8.0 GB of DDR3 RAM onboard on Sun 28 Mar 2010 at 21:44:50 (UTC-0500) EST, but it failed due to hardware problems.

  299,998 Digits of CMRB: The computation began Fri 13 Aug 2010 10:16:20 pm EDT and ended 2.23199*10^6 seconds later |  Wednesday, September 8, 2010. I used Mathematica 6.0 for Microsoft Windows (64-bit) (June 19, 2007) Which is an average of 7.44 seconds per digit. I used my Dell Studio XPS 8100 i7 860 @ 2.80 GHz with 8GB physical DDR3 RAM. Windows 7 reserved an additional 48.929 GB virtual Ram.

300,000 digits to the right of the decimal point of CMRB from Sat 8 Oct 2011 23:50:40 to Sat 5 Nov 2011 19:53:42 (2.405*10^6 seconds later). This run was 0.5766 seconds per digit slower than the 299,998 digit computation even though it used 16 GB physical DDR3 RAM on the same machine. The working precision and accuracy goal combination were maximized for exactly 300,000 digits, and the result was automatically saved as a file instead of just being displayed on the front end. Windows reserved a total of 63 GB of working memory of which 52 GB were recorded being used. The 300,000 digits came from the Mathematica 7.0 command`
    Quit; DateString[]
    digits = 300000; str = OpenWrite[]; SetOptions[str, 
    PageWidth -> 1000]; time = SessionTime[]; Write[str, 
    NSum[(-1)^n*(n^(1/n) - 1), {n, \[Infinity]}, 
    WorkingPrecision -> digits + 3, AccuracyGoal -> digits, 
    Method -> "AlternatingSigns"]]; timeused = 
    SessionTime[] - time; here = Close[str]
    DateString[]

314159 digits of the constant took 3 tries due to hardware failure. Finishing on September 18, 2012, I computed 314159 digits, taking 59 GB of RAM.  The digits came from the Mathematica 8.0.4 code`

    DateString[]
    NSum[(-1)^n*(n^(1/n) - 1), {n, \[Infinity]}, 
    WorkingPrecision -> 314169, Method -> "AlternatingSigns"] // Timing
    DateString[]

1,000,000 digits of CMRB  for the first time in history in 18 days 9 hours 11 minutes 34.253417 seconds by Sam Noble of the Apple Advanced Computation Group.

1,048,576 digits of CMRB in a lightning-fast 76.4 hours, finishing on Dec 11, 2012 were scored by Dr Richard Crandall, an Apple scientist and head of its advanced computational group. That's on a 2.93 GHz 8-core Nehalem.

    To beat that, on Aug of 2018, I computed 1,004,993 digits in 53.5 hours 34 hours computation time (from the timing command) with 10 DDR4 RAM (of up to 3000 MHz) supported processor cores overclocked up to 4.7 GHz! Search this post for "53.5" for documentation. 

    To beat that, on Sept 21, 2018: I computed 1,004,993 digits in 50.37 hours of absolute time and 35.4 hours of computation time (from the timing command) with 18  (DDR3 and DDR4) processor cores!  Search this post for "50.37 hours" for documentation.**

    To beat that, on May 11, 2019, I computed over 1,004,993 digits, in 45.5 hours of absolute time and only 32.5 hours of computation time, using 28 kernels on 18 DDR4 RAM (of up to 3200 MHz) supported cores overclocked up to  5.1 GHz  Search 'Documented in the attached ":3 fastest computers together 3.nb." '  for the post that has the attached documenting notebook.

    To beat that, I accumulated over 1,004,993 correct digits in 44 hours of absolute time and 35.4206 hours of computation time on 10/19/20, using 3/4 of the MRB constant supercomputer 2 -- see https://www.wolframcloud.com/obj/bmmmburns/Published/44%20hour%20million.nb  for documentation.

    To beat that I did a 1,004,993 correct digits computation in 36.7 hours of absolute time and only 26.4 hours of computation time, on Sun 15 May 2022 at 06:10:50, using 3/4  of the MRB constant supercomputer 3. Ram Speed was 4800MHz and all of the 30 cores were clocked at up to 5.2 GHz.

    To beat that I did a 1,004,993 correct digits computation in 35.6 hours of absolute time and only 25.3 hours of computation time, on Wed 3 Aug 2022 08:05:38, using the MRB constant supercomputer 3. Ram Speed was 4000MHz and all of the 40 cores were clocked at up to 5.5 GHz.

44 hours million notebook

36.7 hours million notebook

35.6 hours million notebook

 A little over 1,200,000 digits, previously, of CMRB in 11   days, 21 hours, 17 minutes, and 41 seconds (I finished on March 31, 2013, using a six-core Intel(R) Core(TM) i7-3930K CPU @ 3.20 GHz. see https://www.wolframcloud.com/obj/bmmmburns/Published/36%20hour%20million.nb

for details.

2,000,000 or more digit computation of CMRB on May 17, 2013, using only around 10GB of RAM. It took 37 days 5 hours 6 minutes 47.1870579 seconds. I used my six-core Intel(R) Core(TM) i7-3930K CPU @ 3.20 GHz 3.20 GHz.

 3,014,991 digits of CMRB,  world record computation of **C**<sub>*MRB*</sub> was finished on Sun 21 Sep 2014 at 18:35:06. It took 1 month 27 days 2 hours 45 minutes 15 seconds. The processor time from the 3,000,000+ digit computation was 22 days. I computed the 3,014,991 digits of **C**<sub>*MRB*</sub> with Mathematica 10.0. I Used my new version of Richard Crandall's code in the attached 3M.nb, optimized for my platform and large computations. I also used a six-core Intel(R) Core(TM) i7-3930K CPU @ 3.20 GHz with 64 GB of RAM of which only 16 GB was used. Can you beat it (in more number of digits, less memory used, or less time taken)? This confirms that my previous "2,000,000 or more digit computation" was accurate to 2,009,993 digits. they were used to check the first several digits of this computation. See attached 3M.nb for the full code and digits.

 Over 4 million digits of CMRB was finished on Wed 16 Jan 2019 19:55:20.
It took 4 years of continuous tries. This successful run took 65.13 days absolute time, with a processor time of 25.17 days, on a 3.7 GHz overclocked up to 4.7 GHz on all cores Intel 6 core computer with 3000 MHz RAM. According to this computation, the previous record, 3,000,000+ digit computation, was accurate to 3,014,871 decimals, as this computation used my algorithm for computing n^(1/n) as found in chapter 3 in the paper at 

https://www.sciencedirect.com/science/article/pii/0898122189900242 and the 3 million+ computation used Crandall's algorithm. Both algorithms outperform Newton's method per calculation and iteration.


Example use of M R Burns' algorithm to compute 123456789^(1/123456789) 10,000,000 digits:

ClearSystemCache[]; n = 123456789;
(*n is the n in n^(1/n)*)
x = N[n^(1/n),100];
(*x starts out as a relatively small precision approximation to n^(1/n)*)
pc = Precision[x]; pr = 10000000;
(*pr is the desired precision of your n^(1/n)*)
Print[t0 = Timing[While[pc < pr, pc = Min[4 pc, pr];
x = SetPrecision[x, pc];
y = x^n; z = (n - y)/y;
t = 2 n - 1; t2 = t^2;
x = x*(1 + SetPrecision[4.5, pc] (n - 1)/t2 + (n + 1) z/(2 n t)
- SetPrecision[13.5, pc] n (n - 1)/(3 n t2 + t^3 z))];
(*You get a much faster version of N[n^(1/n),pr]*)
N[n - x^n, 10]](*The error*)];
ClearSystemCache[]; n = 123456789; Print[t1 = Timing[N[n - N[n^(1/n), pr]^n, 10]]]

 Gives

  {25.5469,0.*10^-9999984}

  {101.359,0.*10^-9999984}




  More information is available upon request.

 More than 5 million digits of CMRB were found on Fri 19 Jul 2019 18:49:02,   Methods described in the reply below which begins with "Attempts at a 5,000,000 digit calculation ."   For this 5 million digit calculation of **C**<sub>*MRB*</sub> using the 3 node MRB supercomputer: processor time was 40 days. and the actual time was 64 days.   That is in less absolute time than the 4-million-digit computation which used just one node.

6,000,000 digits of CMRB after 8 tries in 19 months. (Search "8/24/2019 It's time for more digits!" below.) finishing on Tue 30 Mar 2021 at 22:02:49 in 160 days.
    The MRB constant supercomputer 2 said the following:
    Finished on Tue 30 Mar 2021 22:02:49. computation and absolute time were
    5.28815859375*10^6 and 1.38935720536301*10^7 s. respectively
    Enter MRB1 to print 6029991 digits. The error from a 5,000,000 or more-digit calculation that used a different method is      
    0.*10^-5024993.

That means that the 5,000,000-digit computation Was actually accurate to 5024993 decimals!!!

5,609,880, verified by 2 distinct algorithms for x^(1/x), digits of CMRB on Thu 4 Mar 2021 at 08:03:45. The 5,500,000+ digit computation using a totally different method showed that many decimals are in common with the 6,000,000+ digit computation in 160.805 days.

6,500,000 digits of CMRB on my second try,

The MRB constant supercomputer said,

Finished on Wed 16 Mar 2022 02: 02: 10. computation and absolute time
were 6.26628*10^6 and 1.60264035419592*10^7s respectively Enter MRB1
to print 6532491 digits. The error from a 6, 000, 000 or more digit
calculation that used a different method is 
0.*10^-6029992.

"Computation time" 72.526 days

 "Absolute time" 185.491 days

C_MRB and its applications

Definition 1 C_MRB is defined at https://en.wikipedia.org/wiki/MRB_constant .

From Wikipedia:

References
 Plouffe, Simon. "mrburns". Retrieved 12 January 2015.
 Burns, Marvin R. (23 January 1999). "RC". math2.org. Retrieved 5 May 2009.
 Plouffe, Simon (20 November 1999). "Tables of Constants" (PDF). Laboratoire de combinatoire et d'informatique mathématique. Retrieved 5 May 2009.
 Weisstein, Eric W. "MRB Constant". MathWorld.
 Mathar, Richard J. (2009). "Numerical Evaluation of the Oscillatory Integral Over exp(iπx) x^*1/x) Between 1 and Infinity". arXiv:0912.3844 [math.CA].
 Crandall, Richard. "Unified algorithms for polylogarithm, L-series, and zeta variants" (PDF). PSI Press. Archived from the original (PDF) on April 30, 2013. Retrieved 16 January 2015.
 (sequence A037077 in the OEIS)
 (sequence A160755 in the OEIS)
 (sequence A173273 in the OEIS)
 Fiorentini, Mauro. "MRB (costante)". bitman.name (in Italian). Retrieved 14 January 2015.
 Finch, Steven R. (2003). Mathematical Constants. Cambridge, England: Cambridge University Press. p. 450. ISBN 0-521-81805-2.

`

The following equation that was shown in the Wikipedia definition shows how closely the MRB constant is related to root two.

In[1]:= N[Sum[Sqrt[2]^(1/n)* Sqrt[n]^(1/n) - ((Sqrt[2]^y*Sqrt[2]^x)^(1/Sqrt[2]^x))^Sqrt[2]^(-y)/. 
x -> 2*Log2[a^2 + b^2] /. 
y -> 2*Log2[-ai^2 - bi^2] /. 
a -> 1 - (2*n)^(1/4) /. 
b -> 2^(5/8)*Sqrt[n^(1/4)] /. 
ai -> 1 - I*(2*n)^(1/4) /. 
bi -> 2^(5/8)*Sqrt[I*n^(1/4)], {n, 1, Infinity}], 7]

Out[1]= 0.1878596 + 0.*10^-8 I

The complex roots and powers above are found to be well-defined because we get all either "integer" and "rational" the first of the following lists only, also by working from the bottom to the top of the above list of equations.

Code:

In[349]:= Table[
 Head[FullSimplify[
   Expand[(Sqrt[2])^-y/(Sqrt[2])^x] //. 
     x -> 2 (Log[1 + Sqrt[2] Sqrt[n]]/Log[2]) /. 
    y -> 2 (Log[-1 + Sqrt[2] Sqrt[n]]/Log[2])]], {n, 1, 10}]

Out[349]= {Integer, Rational, Rational, Rational, Rational, Rational, \
Rational, Rational, Rational, Rational}

In[369]:= Table[
 Head[FullSimplify[
   Expand[(Sqrt[2])^-y/(Sqrt[2])^x] //. 
     x -> 2 (Log[1 + Sqrt[2] Sqrt[n]]/Log[3]) /. 
    y -> 2 (Log[-1 + Sqrt[2] Sqrt[n]]/Log[2])]], {n, 1, 10}]

Out[369]= {Times, Rational, Times, Times, Times, Times, Times, Times, \
Times, Times}

Definition 2 C_MRB is defined at http://mathworld.wolfram.com/MRBConstant.html.

From MathWorld:

SEE ALSO:
Glaisher-Kinkelin Constant, Power Tower, Steiner's Problem
REFERENCES:
Burns, M. R. "An Alternating Series Involving n^(th) Roots." Unpublished note, 1999.

Burns, M. R. "Try to Beat These MRB Constant Records!" http://community.wolfram.com/groups/-/m/t/366628.

Crandall, R. E. "Unified Algorithms for Polylogarithm, L-Series, and Zeta Variants." 2012a.

http://www.marvinrayburns.com/UniversalTOC25.pdf.

Crandall, R. E. "The MRB Constant." §7.5 in Algorithmic Reflections: Selected Works. PSI Press, pp. 28-29, 2012b.

Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, p. 450, 2003.

Plouffe, S. "MRB Constant." http://pi.lacim.uqam.ca/piDATA/mrburns.txt.

Sloane, N. J. A. Sequences A037077 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha: MRB Constant
CITE THIS AS:
Weisstein, Eric W. "MRB Constant." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/MRBConstant.html

How would we show that the any of the series in the above MathWorld definition are convergent, or even absolutely convergent?

For "a"_k=k^1/k, given that the sequence is monotonically decreasing according to Steiner's Problem, next, we would like to show (5) is the alternating sum of a sequence that converges to 0 monotonically and use the Alternating series test to see that it is conditionally convergent

Here is proof that 1 is the limit of "a" as k goes to infinity:

Here are many other proofs that 1 is the limit of "a" as k goes to infinity.

Thus, (k^1/k-1) is a monotonically decreasing and bounded below by 0 sequence.

If we want an absolutely convergent series, we can use (4).

S_k which, since the sum of the absolute values of the summands is finite, the sum converges absolutely!

There is no closed-form for C_MRB in the MathWorld definition; this could be due to the following: in Mathematical Constants,( Finch, S. R. Mathematical Constants, Cambridge, England: Cambridge University Press, p. 450), Steven Finch wrote that it is difficult to find an "exact formula" (closed-form solution) for it.

Real-World, and beyond, Applications

This section and the rest of the content of this first post were moved below to improve loading times. CNRL+F "Real-World, and beyond, Applications" to finish reading it.