The MRB constant: ALL ABOARD!
POSTED BY: Marvin Ray Burns.
C_{MRB} is defined at
https://en-academic.com/, Wikipedia, Mathematical constant;
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;
Encyclopedia of Mathematics (Series #94);
Engineering Tools of the Iran Civil Center, 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;
OEIS Encyclopedia (The MRB constant);
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.
Content of this first post, as of May 24, 2022
- How it all began
- The why and what of the C_{MRB} Records,
- C_{MRB} and its applications,
- MeijerG Representation for C_{MRB},
- C_{MRB} formulas and identities,
- Primary Proof 1,
- Primary Proof 2,
- Primary Proof 3,
- The relationship between C_{MRB} and its integrated analog,
- 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 an incredible 7 million digits and the MRB constant supercomputer 3.
How it all began
From these meager beginnings:
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 have studied a few graduate-level topics in Mathematics but have yet to earn my Bachelor's, which I intend to work on in the coming year while still working full-time.
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.
I came to find out that this constant (CMRB)
was more closely related to other constants than I could have imagined:
For example, consider its relationship to Viswanath's constant (VC)
With both being functions of x^{1/x} alone, we have these approximations:
In 2016 I found
==3.57*10^-11.
In[232]:= (-801233750 + 459067873 \[Pi] -
827447500 CMRB \[Pi])/134832500 - VC
Out[232]= 3.57*10^-11
The following two much simpler ones, with the memory of Zeta(2) = Pi^2/6 and Pi/2≈11/7, bear some resemblance to a would-be limit-sum analog to the Abel-Plana integral-sum equations :
In[2]:= Reduce[w Exp[w] == 1, w]
Out[2]= C[1] \[Element] Integers && w == ProductLog[C[1], 1]
The Mathematica code and result:
VC = 1.1319882487943061;
CMRB = 0.1878596424620671202485179 ;
VC/(6(ProductLog[E]))-CMRB
(* 0.00080506567032*).
VC/(66/7-6ProductLog[1])-CMRB
(* 3.416388*^-8*).
It is difficult to describe any such analog thus far, though.
I believe they are approximations rather than equations. Nonetheless, they are ratios of integers and named constants.
But wait, there is more to this Viswanath's-MRB constant "relationship:"
and
Here are those codes:
CMRB = NSum[(-1)^n*(n^(1/n) - 1), {n, 1, Infinity},
WorkingPrecision -> 20, Method -> "AlternatingSigns"];
For[n = 1, n < 10,
Print[WolframAlpha[ToString[N[((56/5)*CMRB)^(1/n), 10]],
"PodCells"][[6]]], n++]
and
So, are the division and surd approximations related?
I don't know, but their ratio is approximately Gelfond's constant
$=e^\pi.$
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 it is a very slow to converge ocellary integral that could later be looked up in Google Scholar.
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:
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)
[ 2015 ] [ 2021 Method ]
V 10.1.2 V10.3 v11.3 V12.0 V12.1 V12.3 V13.0
1000 3.3 3.1
2000 437 256 67 67 58 21 20
3000 889 794 217 211 186 84 ?
4000 1633 514 492 447 253* 259 (248*)
5000 2858 1005 925 854 386 378
10000 17678 8327 7748 7470 2800 2748
20000 121,431 71000 66177
40000 362,945 148,817 134,440
* means from a fresh kernel.
From these meager beginnings:
On Sat, 6 Feb 2010, 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?)
$C$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
$a$.
The solution I got surprised me: it was
$a=1-2\times C\mathrm{MRB}=0.6242807150758657595029641318914535398881938101997224\ldots$.
Where
$C\mathrm{MRB}$ is
$\sum_{n=1}^\infty (-1)^n\times(n^{1/n}-1)$.
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.
I wrote Mathematica to find out what method is used and they replied:
From Marvin Ray Burns bmmmburns@sbcglobal.net Date: Sat, 6 Feb 2010
21:03:54 -0600 Subject: Premier Service Help Form To:
support@wolfram.com
Name: Marvin Ray Burns Email: ******.net Organization: IUPUI License: L***** Version: 6 OS: "Windows"
Suggestion or Bug: I am writing about the "AlternatingSigns" Method in NSum. Consider the Divergent series NSum[(-1)^n*(n^(1/n) - a), {n, Infinity}, Method -> "AlternatingSigns"], where Abs[a]<1. Strangely, Mathematica does compute a value for many of those series. Obviously, the algorithms used in the "AlternatingSigns" Method produce a value independently of the fact that the series is divergent. Perhaps one should not be surprised that a value is returned but IT SHOULD BE MADE CLEAR WHAT THAT VALUE REPRESENTS! What is the meaning of that value given that its series is divergent? From my experiments in V7 that value is not reproducible with Sum[, Regularization-> ].`
Hello,
Thank you for the email.
`Note that in V7, you may also try these:b `In[11]:= Sum[(-1)^n*(n^(1/n) - 0.624277766757), {n, Infinity},Method -> "AlternatingSigns"]`b> During evaluation of In[11]:= Sum::div:Sum does not converge. >>b `Out[11]= Sum[(-1)^n*(-0.624277766757 + n^(1/n)), {n, Infinity}, Method -> "AlternatingSigns"]``In[33]:= Sum[(-1)^n*(n^(1/n) - 0.624277766757), {n, Infinity}]` During evaluation of In[33]:= Sum::div:Sum does not converge. >> `Out[33]= Sum[(-1)^n*(-0.624277766757 + n^(1/n)), {n, Infinity}]` Thank you for taking the time to send us this report. I have forwarded your examples to our development group. We apologize for any inconvenience caused by this problem.`
I have included your contact information so that you can be notified
when this has been resolved.
Sincerely,
*******, Ph.D. Technical Support Wolfram Research, Inc. http://support.wolfram.com
`Thank you for your email. If you do not specify a Method for NSum it will try to choose between the EulerMaclaurin or WynnEpsilon methods. In any case, some implicit assumptions about the functions you are summing have to be made. If these assumptions are not correct, you may get inaccurate answers`.
Numerical Sums and Products f the ratio test does not give 1, the Wynn epsilon algorithm is applied to a sequence of partial sums or products. Otherwise Euler[Dash]Maclaurin summation is used with Integrate or NIntegrate.`
1-2$C$MRB is what I call Ma. Here is the formula for Ma by way of Levin-Type Sequence Transformations: in Maple
`Digits := 20; fsolve(sum((-1)^j*(j^(1/j)-a), j = 1 .. infinity) = 0, a)`,
giving 0.62428071507586575950,
and in Mathematica,
Block[{$MaxExtraPrecision = 10000}, FindRoot[NSum[(-1)^j*(j^(1/j) - a), {j, Infinity}], {a, 0.6}, WorkingPrecision -> 20]]
giving {n -> 0.62428071507608096085}.
There is a discrepancy between the two results, but they do agree that Ma is approximately = 0.624280715076.
Finally let a = Ma =
$1-2C$MRB = 0.6242807150758... and the two limit-points of the series
$\sum_{n=1}^\infty (-1)^n(n^{1/n}-Ma)$ are +/-
$C$MRB with its Levin's u-transform's result being 0.
See the integer sequence here.
At about the same time, I noticed the following when I extend the meaning of $\sum$ through "summation methods", whereby series that diverge in one sense converge in another sense (e.g. Cesaro, etc.)
Let
$c=$MRB constant;
$x$, and
$y =$ any number,
then it can be shown that
$$\sum_{n=1}^\infty (-1)^n(x n^{1/n}+y n)=(c-1/2) x-\frac14 y$$
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). 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 mention in a following post, where it is written, "Using those ratios, it looks like" (There m is the MRB constant.)
MRB Constant Records,
My inspiration to compute a lot of digits of CMRB 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 that is the only way to solve such a problem as1465528573348167959709563453947173222018952610559967812891154^ m-m, which gives
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 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 ) CMRB 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][41] if you can read the printed and scanned copy there.
5,000 digits of CMRB in September of 1999:on a 350 MHz Pentium II with 64 Mb of RAM using the simple PARI commands \p 5000;sumalt(n=1,((-1)^n*(n^(1/n)-1))), after allocating enough memory.
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,.
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.
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 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 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 of CMRB 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 of CMRB 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 of CMRB 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][42] for documentation.
To beat that I did a 1,004,993 correct digits computation in 36 hours of absolute time and only 26 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.
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.
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 CMRB 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 CMRB 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 MRB 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 the MRB constant 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 the MRB constant 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.
6,500,000 digits of the MRB constant 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
The MRB constant 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}, gven 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 a 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
CMRB as a Growth Model
Its factor
models the interest rate to multiply an investment k times in k periods, as well as "other growth and decay functions involving the more general expression , as in Plot 1A," because
r=(k^(1/k)-1);Animate[ListPlot[l=Accumulate[Table[(r+1)^n,{k,100}]], PlotStyle->Red,PlotRange->{0,150},PlotLegends->{"\!\(\*UnderscriptBox[\(\[Sum]\), \(\)]\)(r+1\!\(\*SuperscriptBox[\()\), \(n\)]\)/.r->(\!\(\*SuperscriptBox[\(k\), \(1/k\)]\)-1)/.n->"n},AxesOrigin->{0,0}],{n,0,5}]
Plot 1A
The discrete rates looks like the following.
r = (k^(1/k) - 1); me =
Animate[ListPlot[l = Table[(r + 1)^n, {k, 100}], PlotStyle -> Red,
PlotLegends -> {"(r+1)^n/.r->\!\(\*SuperscriptBox[\(k\), \
\(1/k\)]\)=1/.n->", n}, AxesOrigin -> {0, 0},
PlotRange -> {0, 7}], {n, 1, 5}]
That factor models not only discretely compounded rates but continuous too, ie
By entering
Solve[P*E^(r*t) == P*(t^(1/t) - 1), r]
we see, for
gives an effect of continuous decay of Here Q1 means the first Quarter form 0 to -1.
The alternating sum of the principal of those continuous rates, i.e. P=(-1)^{t} e^{r t} is the MRB constant (CMRB):
In[647]:= NSum[(-1)^t ( E^(r*t)) /. r -> Log[-1 + t^(1/t)]/t, {t, 1,
Infinity}, Method -> "AlternatingSigns", WorkingPrecision -> 30]
Out[647]= 0.18785964246206712024857897184
Its integral (MKB) is an analog to C_{MRB} :
In[1]:= NIntegrate[(-1)^t (E^(r*t)) /. r -> Log[-1 + t^(1/t)]/t, {t,
1, Infinity I}, Method -> "Trapezoidal", WorkingPrecision -> 30] -
2 I/Pi
Out[1]= 0.0707760393115288035395280218303 -
0.6840003894379321291827444599927 I
So, integrating P yields about 1/2 greater of a total than summing:
In[663]:=
CMRB = NSum[(-1)^n ( Power[n, ( n)^-1] - 1), {n, 1, Infinity},
Method -> "AlternatingSigns", WorkingPrecision -> 30];
In[664]:=
MKB = Abs[
NIntegrate[(-1)^t ( E^(r*t)) /. r -> Log[-1 + t^(1/t)]/t, {t, 1,
Infinity I}, Method -> "Trapezoidal", WorkingPrecision -> 30] -
2 I/Pi];
In[667]:= MKB - CMRB
Out[667]= 0.49979272646562724956073343752
Next:
CMRB from Geometric Series and Power Series
The MRB constant: is closely related to geometric series:
The inverse function of the "term" of the MRB constant, i.e. x^(1/x) within a certain domain is solved for in this link,
...
Consider the following about a slight generalization of that term.
C_{MRB} can be written in geometric series form:
C_{MRB}=
In[240]:= N[Quiet[(Sum[q^k, {x, 1, Infinity}] /.
k -> Log[-E^(I*Pi*x) + E^(x*(I*Pi + Log[x]/x^2))]/Log[q]) -
Sum[E^(I*Pi*x)*(-1 + x^(1/x)), {x, 1, Infinity}]]]
Out[240]= -4.163336342344337*^-16
Here too, with some shifting:
C_{MRB}=
In[269]:= Quiet[
N[Sum[q^p - 1, {x, 1, Infinity}] -
Sum[E^(I*Pi*x)*(x^(1/x) - 1), {x, 1, Infinity}]] /.
p -> Log[(\!\(TraditionalForm\`
\*SuperscriptBox[\((\(-1\))\), \(x\)] \((\
\*SuperscriptBox[\(q\), \(k\)] - 1)\) + 1\))]/Log[q] /.
k -> Log[x]/(x*Log[q])]
Out[269]= 0.
It is also related to a power series of the form
, shifted by 1,
where we have
C_{MRB}=
Sum[(-1)^x (q^k - 1) /. k -> Log[x]/(x Log[q]), {x, 1, Infinity}]
In[70]:= N[%]
Out[70]= 0.18786
Next
The Geometry of the MRB constant
In 1837 Pierre Wantzel proved that an nth root of a given length cannot be constructed if n is not a power of 2 (as mentioned here in Wikipedia). However, the following is a little different.
For on November 21, 2010, I coined a multiversal analog to, Minkowski space that plots their values from constructions arising from a peculiar non-euclidean geometry, below, and fully in this vixra draft.
As in Diagram 2, we give each n-cube a hyperbolic volume (content) equal to its dimension,
Geometrically, as in Diagram3, on the y,z-plane line up an edge of
each n-cube. The numeric values displayed in the diagram are the partial sums of S[x_] = Sum[(-1)^n*n^(1/n), {n, 1, 2*u}]
where u is an positive integer. Then M is the MRB constant.
Join[ Table[N[S[x]], {u, 1, 4}], {"..."}, {NSum[(-1)^n*(n^(1/n) - 1), {n, 1, Infinity}]}]
Out[421]= {0.414214, 0.386178, 0.354454, 0.330824, "...", 0.18786}
Next
The publishing of CMRB.
Google Scholar results on MRB constant 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.
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.
CMRB is at the 38th most popular Wikipedia Mathematics page as of April 6, 2022. As of 20 April 2022, there are 6,486,890 articles in the English Wikipedia.
The geometry of the MRB constant's connection to string theory describing black holes results found in Ramanujan's modular equations, a connection to the Quantum Cosmological Constant, MRB constant integrals and relationships to other constants in string theory, in a relationship with the Ramanujan-Nardelli mock general formula, from Ramanujan's Mock Theta Functions to Black Hole Entropies and Symmetry, Supersymmetry, Golden Ratio, the MRB Constant and the linked integrals to it, applied to several equations of Geometric Measure Theory, to the Ramanujan’s equations and connections with some sectors of String Theory, new possible mathematical connections between the MRB constant and various sectors of Theoretical Physics and Cosmology, the study of some integrals concerning the MRB Constant and several equations of Geometric Measure Theory, analyzing new possible mathematical connections with some Cosmological parameters and sectors of String Theory, analyzing some Cosmological parameters and sectors of String Theory, and much more
in this link.
Concerning his prolific writing on the MRB constant, 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 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.
https://www.academia.edu/75884771/
https://www.academia.edu/76084911/
https://www.academia.edu/76405749/
https://www.academia.edu/76784160/
https://www.academia.edu/77164290/
https://www.academia.edu/77531870/
https://www.academia.edu/77752950/
https://www.academia.edu/77978967/
https://www.academia.edu/78104771/
https://www.academia.edu/72576179/
https://www.academia.edu/72674127/
https://www.academia.edu/73043410/
https://www.academia.edu/73201689/
MeijerG Representation
From its integrated analog, I found a MeijerG representation for C_{MRB}.
The search for it began with the following:
On 10/10/2021, I found the following proper definite integral that leads to almost identical
proper integrals from 0 to 1 for C_{MRB} and its integrated analog.
See notebook in this link.
Here is a MeijerG function for the integrated analog. See (proof) of discovery.
f(n)=.
`
In[135]:=f[n_]:=MeijerG[{{},Table[1,{n+1}]},{Prepend[Table[0,n+1],-n+1],{}},-\[ImaginaryI]\[Pi]];`
In[337]:=M2=NIntegrate[E^(I Pi x)(SuperscriptBox["x", FractionBox["1", "x"]]-1),
{x,1,Infinity I},WorkingPrecision->100]
Out[337]=0.07077603931152880353952802183028200136575469620336302758317278816361845726438203658083188126617723821-0.04738061707035078610720940650260367857315289969317363933196100090256586758807049779050462314770913485 \[ImaginaryI]
I wonder if there is one for the MRB constant sum (CMRB)?
According to "Primary Proof 1" and "Primary Proof 3" shown below along with the section prefixed by the phrase "So far I came up with,"
it can be proven that for
G being the Wolfram MeijerG function
and f(n)=, and
g[x_] = (-1)^x (1 - (x + 1)^(1/(x + 1)));
In[52]:= (1/2)*
NIntegrate[(g[-t] - g[t])/(Sin[Pi*t]*Cos[Pi*t]*I + Sin[Pi*t]^2), {t,
0, I*Infinity}, WorkingPrecision -> 100,
Method -> "GlobalAdaptive"]
Out[52]= 0.\
1170836031505383167089899122239912286901483986967757585888318959258587\
7430027817712246477316693025869 +
0.0473806170703507861072094065026036785731528996931736393319610009025\
6586758807049779050462314770913485 I
In[57]:= Re[
NIntegrate[
g[-t]/(Sin[Pi*t]*Cos[Pi*t]*I + Sin[Pi*t]^2), {t, 0, I*Infinity},
WorkingPrecision -> 100,
Method -> "GlobalAdaptive"]]
Out[57]= 0.\
1878596424620671202485179340542732300559030949001387861720046840894772\
315646602137032966544331074969
MRB constant formulas and identities
I developed this informal catalog of formulas for the MRB constant with over 20 years of research and ideas from users like you.
3/25/2022
Formula (11) =
As Matheamatica says:
Assuming[Element[c, \[DoubleStruckCapitalZ]], FullSimplify[
E^(t*(r + I*Pi*(2*c + 1))) /. r -> Log[t^(1/t) - 1]/t]]
= E^(I (1 + 2 c) [Pi] t) (-1 + t^(1/t))
Where for all integers c, (1+2c) is odd leading to
Expanding the E^log term gives
which is ,
That is exactly (2) in the above-quoted MathWorld definition:
2/21/2022
Directly from the formula of 12/29/2021 below,
In
u = (-1)^t; N[
NSum[(t^(1/t) - 1) u, {t, 1, Infinity }, WorkingPrecision -> 24,
Method -> "AlternatingSigns"], 15]
Out[276]= 0.187859642462067
In
v = (-1)^-t - (-1)^t; 2 I N[
NIntegrate[Im[(t^(1/t) - 1) v^-1], {t, 1, Infinity I},
WorkingPrecision -> 24], 15]
Out[278]= 0.187859642462067
Likewise,
Expanding the exponents,
This can be generalized to
Building upon that, we get a closed form for the inner integral in the following.
In[1]:=
CMRB = NSum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity},
WorkingPrecision -> 1000, Method -> "AlternatingSigns"];
In[2]:= CMRB - {
Quiet[Im[NIntegrate[
Integrate[
E^(Log[t]/t + x)/(-E^((-I)*Pi*t + x) + E^(I*Pi*t + x)), {x,
I, -I}], {t, 1, Infinity I}, WorkingPrecision -> 200,
Method -> "Trapezoidal"]]];
Quiet[Im[NIntegrate[
Integrate[
Im[E^(Log[t]/t + x)/(-E^((-I)*Pi*t + x) + E^(I*Pi*t + x))], {x,
-t, t }], {t, 1
, Infinity I}, WorkingPrecision -> 2000,
Method -> "Trapezoidal"]]]}
Out[2]= {3.*10^-998, 3.*10^-998}
Which after a little analysis, can be shown convergent in the continuum limit at t → ∞ i.
12/29/2021
From "Primary Proof 1" worked below, it can be shown that
Mathematica knows that because
m = N[NSum[-E^(I*Pi*t) + E^(I*Pi*t)*t^t^(-1), {t, 1, Infinity},
Method -> "AlternatingSigns", WorkingPrecision -> 27], 18];
Print[{m -
N[NIntegrate[
Im[(E^(Log[t]/t) + E^(Log[t]/t))/(E^(I \[Pi] t) -
E^(-I \[Pi] t))] I, {t, 1, -Infinity I},
WorkingPrecision -> 20], 18],
m - N[NIntegrate[
Im[(E^(Log[t]/t) + E^(Log[t]/t))/(E^(-I \[Pi] t) -
E^(I \[Pi] t))] I, {t, 1, Infinity I},
WorkingPrecision -> 20], 18],
m + 2 I*NIntegrate[
Im[(E^(I*Pi*t + Log[t]/t))/(-1 + E^((2*I)*Pi*t))], {t, 1,
Infinity I}, WorkingPrecision -> 20]}]
yields
{0.*^-19,0.*^-19,0.*^-19}
Partial sums to an upper limit of (10^n i) give approximations for the MRB constant + the same approximation *10^-(n+1) i.
Example:
-2 I*NIntegrate[
Im[(E^(I*Pi*t + Log[t]/t))/(-1 + E^((2*I)*Pi*t))], {t, 1, 10^7 I},
WorkingPrecision -> 20]
gives
0.18785602000738908694 + 1.878560200074*10^-8 I
where CMRB ≈ 0.187856.
Notice it is special because if we integrate only the numerator, we have MKB=, which defines the "integrated analog of C_{MRB}" (MKB) described by Richard Mathar in https://arxiv.org/abs/0912.3844. (He called it M1.)
Like how this:
NIntegrate[(E^(I*Pi*t + Log[t]/t)), {t, 1, Infinity I},
WorkingPrecision -> 20] - I/Pi
converges to
0.070776039311528802981 - 0.68400038943793212890 I.
(The upper limits " i infinity" and " infinity" produce the same result in this integral.)
11/14/2021
Here is a standard notation for the above mentioned
C_{MRB,}
.
In[16]:= CMRB = 0.18785964246206712024851793405427323005590332204; \
CMRB - NSum[(Sum[
E^(I \[Pi] x) Log[x]^n/(n! x^n), {x, 1, Infinity}]), {n, 1, 20},
WorkingPrecision -> 50]
Out[16]= -5.8542798212228838*10^-30
In[8]:= c1 =
Activate[Limit[(-1)^m/m! Derivative[m][DirichletEta][x] /. m -> 1,
x -> 1]]
Out[8]= 1/2 Log[2] (-2 EulerGamma + Log[2])
In[14]:= CMRB -
N[-(c1 + Sum[(-1)^m/m! Derivative[m][DirichletEta][m], {m, 2, 20}]),
30]
Out[14]= -6.*10^-30
11/01/2021
: The catalog now appears complete, and can all be proven through Primary Proof 1, and the one with the eta function, Primary Proof 2, both found below.
a ≠b
g[x_] = x^(1/x); CMRB =
NSum[(-1)^k (g[k] - 1), {k, 1, Infinity}, WorkingPrecision -> 100,
Method -> "AlternatingSigns"]; a = -Infinity I; b = Infinity I;
g[x_] = x^(1/x); (v = t/(1 + t + t I);
Print[CMRB - (-I /2 NIntegrate[ Re[v^-v Csc[Pi/v]]/ (t^2), {t, a, b},
WorkingPrecision -> 100])]); Clear[a, b]
-9.3472*10^-94
Thus, we find
here, and
next:
In[93]:= CMRB =
NSum[Cos[Pi n] (n^(1/n) - 1), {n, 1, Infinity},
Method -> "AlternatingSigns", WorkingPrecision -> 100]; Table[
CMRB - (1/2 +
NIntegrate[
Im[(t^(1/t) - t^(2 n))] (-Csc[\[Pi] t]), {t, 1, Infinity I},
WorkingPrecision -> 100, Method -> "Trapezoidal"]), {n, 1, 5}]
Out[93]= {-9.3472*10^-94, -9.3473*10^-94, -9.3474*10^-94, \
-9.3476*10^-94, -9.3477*10^-94}
CNT+F "The following is a way to compute the" for more evidence
For such n, converges to 1/2+0i.
(How I came across all of those and more example code follow in various replies.)
On 10/18/2021
, I found the following triad of pairs of integrals summed from -complex infinity to +complex infinity.
You can see it worked in this link here.
In[1]:= n = {1, 25.6566540351058628559907};
In[2]:= g[x_] = x^(n/x);
-1/2 Im[N[
NIntegrate[(g[(1 - t)])/(Sin[\[Pi] t]), {t, -Infinity I,
Infinity I}, WorkingPrecision -> 60], 20]]
Out[3]= {0.18785964246206712025, 0.18785964246206712025}
In[4]:= g[x_] = x^(n/x);
1/2 Im[N[NIntegrate[(g[(1 + t)])/(Sin[\[Pi] t]), {t, -Infinity I,
Infinity I}, WorkingPrecision -> 60], 20]]
Out[5]= {0.18785964246206712025, 0.18785964246206712025}
In[6]:= g[x_] = x^(n/x);
1/4 Im[N[NIntegrate[(g[(1 + t)] - (g[(1 - t)]))/(Sin[\[Pi] t]), {t, -Infinity I,
Infinity I}, WorkingPrecision -> 60], 20]]
Out[7]= {0.18785964246206712025, 0.18785964246206712025}
Therefore, bringing
back to mind, we joyfully find,
In[1]:= n =
25.65665403510586285599072933607445153794770546058072048626118194900\
97321718621288009944007124739159792146480733342667`100.;
g[x_] = {x^(1/x), x^(n/x)};
CMRB = NSum[(-1)^k (k^(1/k) - 1), {k, 1, Infinity},
WorkingPrecision -> 100, Method -> "AlternatingSigns"];
Print[CMRB -
NIntegrate[Im[g[(1 + I t)]/Sinh[\[Pi] t]], {t, 0, Infinity},
WorkingPrecision -> 100], u = (-1 + t); v = t/u;
CMRB - NIntegrate[Im[g[(1 + I v)]/(Sinh[\[Pi] v] u^2)], {t, 0, 1},
WorkingPrecision -> 100],
CMRB - NIntegrate[Im[g[(1 - I v)]/(Sinh[-\[Pi] v] u^2)], {t, 0, 1},
WorkingPrecision -> 100]]
During evaluation of In[1]:= {-9.3472*10^-94,-9.3472*10^-94}{-9.3472*10^-94,-9.3472*10^-94}{-9.3472*10^-94,-9.3472*10^-94}
In[23]:= Quiet[
NIntegrate[
Im[g[(1 + I t)]/Sinh[\[Pi] t] -
g[(1 + I v)]/(Sinh[\[Pi] v] u^2)], {t, 1, Infinity},
WorkingPrecision -> 100]]
Out[23]= -3.\
9317890831820506378791034479406121284684487483182042179057328100219696\
20202464096600592983999731376*10^-55
In[21]:= Quiet[
NIntegrate[
Im[g[(1 + I t)]/Sinh[\[Pi] t] -
g[(1 - I v)]/(Sinh[-\[Pi] v] u^2)], {t, 1, Infinity},
WorkingPrecision -> 100]]
Out[21]= -3.\
9317890831820506378791034479406121284684487483182042179057381396998279\
83065832972052160228141179706*10^-55
In[25]:= Quiet[
NIntegrate[
Im[g[(1 + I t)]/Sinh[\[Pi] t] +
g[(1 + I v)]/(Sinh[-\[Pi] v] u^2)], {t, 1, Infinity},
WorkingPrecision -> 100]]
Out[25]= -3.\
9317890831820506378791034479406121284684487483182042179057328100219696\
20202464096600592983999731376*10^-55
On 9/29/2021
I found the following equation for C_{MRB} (great for integer arithmetic because
(1-1/n)^k=(n-1)^k/n^k. )
So, using only integers, and sufficiently large ones in place of infinity, we can use
See
In[1]:= Timing[m=NSum[(-1)^n (n^(1/n)-1),{n,1,Infinity},WorkingPrecision->200,Method->"AlternatingSigns"]][[1]]
Out[1]= 0.086374
In[2]:= Timing[m-NSum[(-1)^n/x! (Sum[((-1 + n)^k) /(k n^(1 + k)), {k, 1, Infinity}])^ x, {n, 2, Infinity}, {x, 1,100}, Method -> "AlternatingSigns", WorkingPrecision -> 200, NSumTerms -> 100]]
Out[2]= {17.8915,-2.2*^-197}
It is very much slower, but it can give a rational approximation (p/q), like in the following.
In[3]:= mt=Sum[(-1)^n/x! (Sum[((-1 + n)^k) /(k n^(1 + k)), {k, 1,500}])^ x, {n, 2,500}, {x, 6}];
In[4]:= N[m-mt]
Out[4]= -0.00602661
In[5]:= Head[mt]
Out[5]= Rational
Compared to the NSum formula for m, we see
In[6]:= Head[m]
Out[6]= Real
On 9/19/2021
I found the following quality of C_{MRB}.
On 9/5/2021
I added the following MRB constant integral over an unusual range.
See proof in this link here.
On Pi Day, 2021, 2:40 pm EST,
I added a new MRB constant integral.
We see many more integrals for C_{MRB}.
We can expand
into the following.
xx = 25.65665403510586285599072933607445153794770546058072048626118194\
90097321718621288009944007124739159792146480733342667`100.;
g[x_] = x^(xx/
x); I NIntegrate[(g[(-t I + 1)] - g[(t I + 1)])/(Exp[Pi t] -
Exp[-Pi t]), {t, 0, Infinity}, WorkingPrecision -> 100]
(*
0.18785964246206712024851793405427323005590309490013878617200468408947\
72315646602137032966544331074969.*)
Expanding upon the previously mentioned
we get the following set of formulas that all equal C_{MRB}:
Let
x= 25.656654035105862855990729 ...
along with the following constants (approximate values given)
{u = -3.20528124009334715662802858},
{u = -1.975955817063408761652299},
{u = -1.028853359952178482391753},
{u = 0.0233205964164237996087020},
{u = 1.0288510656792879404912390},
{u = 1.9759300365560440110320579},
{u = 3.3776887945654916860102506},
{u = 4.2186640662797203304551583} or
$
u = \infty .$
Another set follows.
let
x = 1 and
along with the following {approximations}
{u = 2.451894470180356539050514},
{u = 1.333754341654332447320456} or
$
u = \infty $
then
See
this notebook from the wolfram cloud
for justification.
2020 and before:
Also, in terms of the Euler-Riemann zeta function,
C_{MRB} =
Furthermore, as ,
according to user90369 at Stack Exchange, C_{MRB} can be written as the sum of zeta derivatives similar to the eta derivatives discovered by Crandall.
Information about η^{(j)}(k) please see e.g. this link here, formulas (11)+(16)+(19).
In the light of the parts above, where
C_{MRB}
=
=
=
as well as
an internet scholar going by the moniker "Dark Malthorp" wrote:
Primary Proof 1
C_{MRB}=, based on
C_{MRB}
is proven below by an internet scholar going by the moniker "Dark Malthorp."
Primary Proof 2
denoting the kth derivative of the Dirichlet eta function of k and 0 respectively,
was first discovered in 2012 by Richard Crandall of Apple Computer.
The left half is proven below by Gottfried Helms and it is proven more rigorouslyconsidering the conditionally convergent sum, below that. Then the right half is a Taylor expansion of eta(s) around s = 0.
At
https://math.stackexchange.com/questions/1673886/is-there-a-more-rigorous-way-to-show-these-two-sums-are-exactly-equal,
it has been noted that "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.)"
Primary Proof 3
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/?.
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
I wondered about the relationship between CMRB and its integrated analog and asked the following.
So far I came up with
Another relationship between the sum and integral that remains more unproven than I would like is
f[x_] = E^(I \[Pi] x) (1 - (1 + x)^(1/(1 + x)));
CMRB = NSum[f[n], {n, 0, Infinity}, WorkingPrecision -> 30,
Method -> "AlternatingSigns"];
M2 = NIntegrate[f[t], {t, 0, Infinity I}, WorkingPrecision -> 50];
part = NIntegrate[(Im[2 f[(-t)]] + (f[(-t)] - f[(t)]))/(-1 +
E^(-2 I \[Pi] t)), {t, 0, Infinity I}, WorkingPrecision -> 50];
CMRB (1 - I) - (M2 - part)
gives
6.10377910^-23 - 6.10377910^-23 I.
Where the integral does not converge, but Mathematica can give it a value:
Here is my mini cluster of the fastest 3 computers (the MRB constant supercomputer 0) mentioned below:
The one to the left is my custom-built extreme edition 6 core and later with an 8 core 3.4 GHz Xeon processor with 64 GB 1666 MHz RAM..
The one in the center is my fast little 4 core Asus with 2400 MHz RAM.
Then the one on the right is my fastest -- a Digital Storm 6 core overclocked to 4.7 GHz on all cores and with 3000 MHz RAM.