Real-World, and beyond, Applications
CMRB as a Growth Model
Its factor ![a][112]
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 ![(1+k)^n][113], as in Plot 1A," because ![enter image description here][114]
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
![enter image description here][115]
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}]
![enter image description here][116]
That factor ![enter image description here][117] models not only discretely compounded rates but continuous too, ie ![Pt=p0e^rt.][118]
By entering
Solve[P*E^(r*t) == P*(t^(1/t) - 1), r]
we see, for ![Pt=p0e^rt,][119] ![t>e][120]
gives an effect of continuous decay of ![enter image description here][121] 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):
![enter image description here][122]
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} :
![enter image description here][123]
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: ![enter image description here][124] is closely related to geometric series: ![enter image description here][125]
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,][126]
![enter image description here][127]
...
![enter image description here][128]
Now we have the following for the orientated area, from 0 to 1, between the graph of that term and the axis.
![enter image description here][129]
In[344]:= f[x_] = x^(1/x);
In[346]:= CMRB =
NSum[(-1)^x (f[x] - 1), {x, 1, Infinity}, WorkingPrecision -> 20]
Out[346]= 0.18785964246207
In[350]:= (10 (CMRB + 3))/(3 (3 CMRB - 17)) -
NIntegrate[g = -x /. Solve[y == f[x], x], {y, 0, 1},
WorkingPrecision -> 20]
Out[350]= {1.5605*10^-11}
Consider the following about a slight generalization of that term.
![enter image description here][130]
C_{MRB} can be written in geometric series form:
C_{MRB}=
![enter image description here][131]
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
Why would we express CMRB so?
I'm not entirely sure, but we do have the following interestingly intricate graphs that go towards the value of the MRB constant and the MRB constant-1 as the input gets large.
![enter image description here][132]
![enter image description here][133]
![enter image description here][134]
![enter image description here][135]
![enter image description here][136]
![enter image description here][137]
see notebook [here.][138]
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][139] in Wikipedia). However, the following is a little different.
For ![,][140] on November 21, 2010, I coined a multiversal [analog][141] to, [Minkowski space][142] that plots their values from constructions arising from a peculiar non-euclidean geometry, below, and fully in [this vixra draft][143].
As in Diagram 2, we give each n-cube a hyperbolic volume (content) equal to its dimension,![enter image description here][144]
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.
![enter image description here][145]
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}
Here are views of some regions of the plot of a definite integral equal to CMRB..
![enter image description here][146]
In[66]:= Csch[Pi t] Im[(1 + I t)^(1/(1 + I t))]
Out[66]= Csch[[Pi] t] Im[(1 + I t)^(1/(1 + I t))]
In[67]:= f[t_] = Csch[Pi t] Im[(1 + I t)^(1/(1 + I t))]
Out[67]= Csch[[Pi] t] Im[(1 + I t)^(1/(1 + I t))]
ReImPlot[Im[(1 + I t)^(1/(1 + I t))], {t, 0, 1}, PlotStyle -> Blue,
PlotLabels -> {Placed[(1 + I t)^(1/(1 + I t)), Above]}]
![enter image description here][147]
ReImPlot[Im[(1 + I t)^(1/(1 + I t))], {t, 0, 5}, PlotStyle -> Blue,
PlotLabels -> {Placed[(1 + I t)^(1/(1 + I t)), Above]}]
![enter image description here][148]
Show[ReImPlot[Csch[\[Pi] t], {t, 0, 1}, PlotStyle -> Yellow,
PlotLabels -> "Expressions"]]
![enter image description here][149]
Show[ReImPlot[Csch[\[Pi] t], {t, 0, 5}, PlotStyle -> Yellow,
PlotLabels -> "Expressions"]]
![enter image description here][150]
ReImPlot[f[t], {t, 0, 1},
PlotLabel -> NIntegrate[f[t], {t, 0, 1}, WorkingPrecision -> 20],
PlotStyle -> Green, PlotLabels -> "Expressions"]
![enter image description here][151]
ReImPlot[f[t], {t, 0, 5},
PlotLabel -> NIntegrate[f[t], {t, 0, 5}, WorkingPrecision -> 20],
PlotStyle -> Green, PlotLabels -> "Expressions"]
![enter image description here][152]
Next
MeijerG Representation
From its integrated analog, I found a [MeijerG][153] 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.
![m vs m2 0 to 1][154]
See [notebook in this link][155].
Here is a [MeijerG][156] function for the integrated analog. See [(proof)][157] of discovery.
![enter image description here][158]
f(n)=![enter image description here][159].
`
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]
![enter image description here][160]
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)=![enter image description here][161], and
![enter image description here][162]
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
The Laplace transform analogy to the CMRB
see notebook
Likewise, Wolfram Alpha here says
It also adds
Interestingly,
That has the same argument,
,
as the MeijerG transformation of CMRB.
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.
6/7/2022
CMRB
=![enter image description here][163]
=![enter image description here][164]
=![enter image description here][165]
=![enter image description here][166]
=![enter image description here][167]
=![enter image description here][168]
So, using induction, we have.
![enter image description here][169]
![enter image description here][170]
Sum[Sum[(-1)^(x + n), {n, 1, 5}] + (-1)^(x) x^(1/x), {x, 2, Infinity}]
3/25/2022
Formula (11) =
![enter image description here][171]
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))
![enter image description here][172]
Where for all integers c, (1+2c) is odd leading to ![enter image description here][173]
Expanding the E^log term gives
![enter image description here][174]
which is ![enter image description here][175],
That is exactly (2) in the above-quoted MathWorld definition:
![enter image description here][176]
2/21/2022
Directly from the formula of 12/29/2021 below,
![enter image description here][177]
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,
![enter image description here][178]
Expanding the exponents,
![enter image description here][179]
This can be generalized to
![(x+log/][180]
Building upon that, we get a closed form for the inner integral in the following.
CMRB=
![enter image description here][181]
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
![enter image description here][182]
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=![enter image description here][183], which defines the "integrated analog of C_{MRB}" (MKB) described by Richard Mathar in [https://arxiv.org/abs/0912.3844][184]. (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,}![enter image description here][185]
![enter image description here][186].
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
![enter image description here][187] ![enter image description here][188]
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
![enter image description here][189]
[here,][190] and
![enter image description here][191]
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, ![enter image description here][192] 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.
![CMRB= -complex infinity to +complex infinity][193]
You can see it worked [ in this link here][194].
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
![enter image description here][195]
back to mind, we joyfully find,
![CMRB n and 1][196]
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. )
![CMRB integers 1][197]
So, using only integers, and sufficiently large ones in place of infinity, we can use
![CMRB integers 2][198]
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}.
![replace constants for CMRB][199]
On 9/5/2021
I added the following MRB constant integral over an unusual range.
![strange][200]
See proof [in this link here][201].
On Pi Day, 2021, 2:40 pm EST,
I added a new MRB constant integral.
![CMRB][202] ![=][203] ![integral to sum][204]
We see many more integrals for C_{MRB}.
We can expand
![1/x][205]
into the following.
![xx = 25.656654035][206]
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
![enMRB sinh][207]
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
![enter image description here][208]
See
[this notebook from the wolfram cloud][209]
for justification.
2020 and before:
Also, in terms of the Euler-Riemann zeta function,
C_{MRB} =![enter image description here][210]
Furthermore, as ![enter image description here][211],
according to [user90369][212] at Stack Exchange, C_{MRB} can be written as the sum of zeta derivatives similar to the eta derivatives discovered by Crandall.
![zeta hint ][213] Information about η^{(j)}(k) please see e.g. [this link here][214], formulas (11)+(16)+(19).![credit][215]
In the light of the parts above, where
C_{MRB}
= ![k^(1/k)-1][216]
= ![eta'(k)][217]
= ![sum from 0][218] ![enter image description here][219]
as well as ![double equals RHS][220]
an internet scholar going by the moniker "Dark Malthorp" wrote:
![eta *z^k][221]
Primary Proof 1
C_{MRB}=![enter image description here][222], based on
C_{MRB}
![eta equals][223]
![enter image description here][224]
is proven below by an internet scholar going by the moniker "Dark Malthorp."
![Dark Marthorp's proof][225]
Primary Proof 2
![eta sums][226] 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 rigorously![(][227]considering the conditionally convergent sum,![enter image description here][228]![)][229] below that. Then the right half is a Taylor expansion of eta(s) around s = 0.
![n^(1/n)-1][230]
At
[https://math.stackexchange.com/questions/1673886/is-there-a-more-rigorous-way-to-show-these-two-sums-are-exactly-equal][231],
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][232]), 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][233], culminating with the Fubini theorem which is essentially the manipulation Helms is using.)"
![argument 1][234] ![argument 2][235]
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=![hypothesis][236]
Which is the same as
![enter image description here][237]
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][238] .
![Iimofg->1][239]
![Cauchy's Integral Theorem][240]
![Lim surface h gamma r=0][241]
![Lim surface h beta r=0][242]
![limit to 2n-1][243]
![limit to 2n-][244]
Plugging in equations [5] and [6] into equation [2] gives us:
![left][245]![right][246]
Now take the limit as N?? and apply equations [3] and [4] :
![QED][247]
He went on to note that
![enter image description here][248]
I wondered about the relationship between CMRB and its integrated analog and asked the following.
![enter image description here][249]
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:
Update 2015
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.
see notebook
Likewise, Wolfram Alpha here says
It also adds
Interestingly,
That has the same argument,
,
as the MeijerG transformation of CMRB.