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Crandall's eta formula finally proven!

Since November 2012 I've been asking for a proof of Richard Crandall's MRB constant eta algorithm (the second equation below):

enter image description here

Colleague Gottfried Helms wrote the following, easy to understand, two dimensional proof:

enter image description here @

You can see the bottom row in action by entering,

Limit[DirichletEta'[x], x -> 1] - 
 N[Sum[(-1)^n (D[DirichletEta[x], {x, n}]/x! /. x -> n), {n, 2, 50}], 
  60]

Using the 2-D proof we can relativity easily calculate arbitrary digits of the MRB constant using Crandall's eta method:

    m = 
   NSum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity}, 
    Method -> "AlternatingSigns", WorkingPrecision -> 200]

  (*
  0.18785964246206712024851793405427323005590309490013878617200468408947\
  7231564660213703296654433107496903842345856258019061231370094759226630\
  438929348896184120837336626081613602738126379373435283224982*)

 meta = 
 NSum[(-1)^x (Log[x]/x)^n/n!, {x, 1, Infinity}, {n, 1, Infinity}, 
  Method -> "AlternatingSigns", WorkingPrecision -> 200, 
  PrecisionGoal -> 200, AccuracyGoal -> 200, NSumTerms -> 300]

 (*
0.18785964246206712024851793405427323005590309490013878617200468408947\
7231564660213703296654433107496903842345856258019061231370094759226630\
4389293488961841208373366260816136027381263793734352832125528*)

 m - meta

 (* 1.2429*10^-194*)

See http://math.stackexchange.com/questions/549503/mrb-constant-proofs-wanted for conversation.

Also since

FullSimplify[(PolyLog[1, -x]/(x + 1))^n/
   n! == (-1)^n (Log[x + 1]/(x + 1))^n/n!, 
 Element[x, Integers] && Element[n, Integers] && x >= 0 && n >= 0]

gives true,

we can compute m by the following polylog formula

`p = 400; -NSum[(-1)^x (-PolyLog[1, -x]/(x + 1))^n/n!, {x, 1, 
 Infinity}, {n, 1, Floor[p/2]}, Method -> "AlternatingSigns", 
WorkingPrecision -> p, PrecisionGoal -> p, AccuracyGoal -> Infinity,
NSumTerms -> p]

giving`.

0.18785964246206712024851793405427323005590309490013878617200468408947\
7231564660213703296654433107496903842345856258019061231370094759226630\
4389293488961841208373366260816136027381263793734352832125527639621714\
8932170207628206217151671540841268044836354167199851976802527598938993\
9144579835055613509648521071207844423095868129497688526949564204255586\
48367044104252795247106066609263397483410311578168
POSTED BY: Marvin Ray Burns
8 Replies
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The second formula has been proved, but Crandall didn't show it, or how the terms simplify, to me. enter image description here

That is in the following discussion. http://math.stackexchange.com/questions/1673886/is-there-a-more-rigorous-way-to-show-these-two-sums-are-exactly-equal I would like to see anybody improve on the second formula, (sum of c, as the first sum is proven above), in the way Crandall hoped! My efforts in using the sum of c's to compute MRB are archived in my never ending, often visited discussion, http://community.wolfram.com/groups/-/m/t/366628 . See the end, where I began to ask for some help. .
POSTED BY: Marvin Ray Burns

Here is my fastest program for calculating digits of the MRB constant via the eta formula,

enter image description here This program is nearly twice as fast as the one in the first reply in this blog.

prec = 1500;
to = SessionTime[]; 
etaMM[m_, pr_] := 
 Module[{a, d, s, k, b, c}, a[j_] := N[(Log[j + 1]/(j + 1))^m, pr];
  n = Floor[132 /100 pr];
  d = N[(1/2)*(3 + 2*Sqrt[2])^n*(1 + (3 + 2*Sqrt[2])^(-2*n)), prec];
  {b, c, s} = {-1, -d, 0};
  Do[c = b - c;
   s = s + c a[k];
   b = (k + n) (k - n) b/((k + 1) (k + 1/2)), {k, 0, n - 1}];
  Return[N[s/d, pr] (-1)^m]];
eta1 = N[EulerGamma Log[2] - Log[2]^2/2, prec];
MRBtest = 
  eta1 - ParallelSum[(Cos[Pi m]) etaMM[m, prec]/
      N[Gamma[m + 1], prec], {m, 2, Floor[.40 prec]}];
Print[MRBtest]; SessionTime[] - to

gives

(*    During evaluation of In[188]:= 0.18785964246206712024851793405427323005590309490013878617200468408947723156466021370329665443310749690384234585625801906123137009475922663043892934889618412083733662608161360273812637937343528321255276396217148932170207628206217151671540841268044836354167199851976802527598938993914457983505561350964852107120784442309586812949768852694956420425558648367044104252795247106066609263397483410311578167864166891546003422225883800254553968929471142122189105098328712277308020036445215390536395055332203470627551159812828039510219264914673176293516190659816018664245824950697203381992958420935515162514399357600764593291281451709082424915883204169066409334435914806705564692806787007028115009380606938139385953360657987405562062348704329360737819564603104763950664893061360645528067515193508280837376719296866398103094949637496277383049846324563479311575300289212523291816195626973697074865765476071178017195787368300965902260668753656305516567361288150201438756136686552210674305370591039735756191489093690777983203551193362404637253494105428363699717024418551654837279358822008134480961058802030647819619596953756287834812334976385863010140727252923014723333362509185840248037040488819676767601198581116791693527968520441600270861372286889451015102919988536905728659287086875425492533794395347589703563313440382638887986656195980733514739902565778133172261076127975852722742777308985774922305970962572562718836755752978879253616876739403543214513627725492293131262764357321446216187786377154205423128223

    Out[192]= 105.7159526*)
POSTED BY: Marvin Ray Burns

I've been experimenting with the convergence acceleration of alternating series of the eta formula,

enter image description here , where B is the MRB constant and eta^[m] is the m'th derivative DirichletEta function.

I came up with a couple of programs:

One called m1, which with an call the the Sum command gives an approximation of the MRB constant.

m1[u_, p_] := 
 Module[{s, n, d, a, b, c}, n = Floor[1.5 p]; d = (3 + Sqrt[8])^n;
  d = 1/2*(d + 1/d);
  {b, c, s} = {-1, -d, 0};
  Do[c = b - c;
   a = (Log[k + 1]/(k + 1))^u/u!;
   s = s + c*a;
   b = (k + n)*(k - n)*b/((k + 1)*(k + 1/2)), {k, 0, n}];
  (N[-s/d, n/1.5])]

followed by

p = 100; Sum[m1[u, p], {u, 1, Floor[p/1.8]}]

gives 100 or p digits.

The second one, called meta1, gives an arithmetic sequence for exactly, or more than, p correct digits of the the alternating sum of "ets" eta derivatives shown here:

enter image description here

m2[u_, p_] := 
 Module[{s, n, d, a, b, c}, n = Floor[1.5 p]; d = Cos[n ArcCos[3]];
  {b, c, s} = {-1, -d, 0};
  Do[c = b - c;
   a = (Log[k + 1]/(k + 1))^u/u!;
   s = s + c*a;
   b = (k + n)*(k - n)*b/((k + 1)*(k + 1/2)), {k, 0, n}];
  (-s/d)]

followed by

 ets = 1; p = 10; meta1 = Sum[m2[u, p], {u, 1, ets}]

, gives the new, "trig-log," sequence that follows.

I

 (-(1/2) (451 - Cos[15 ArcCos[3]]) Log[2] - 
   1/3 (-34051 + Cos[15 ArcCos[3]]) Log[3] - 
   1/4 (1024131 - Cos[15 ArcCos[3]]) Log[4] - 
   1/5 (-16299651 + Cos[15 ArcCos[3]]) Log[5] - 
   1/6 (158192259 - Cos[15 ArcCos[3]]) Log[6] - 
   1/7 (-1018147459 + Cos[15 ArcCos[3]]) Log[7] - 
   1/8 (4590269059 - Cos[15 ArcCos[3]]) Log[8] - 
   1/9 (-15068492419 + Cos[15 ArcCos[3]]) Log[9] - 
   1/10 (37120701059 - Cos[15 ArcCos[3]]) Log[10] - 
   1/11 (-70547206787 + Cos[15 ArcCos[3]]) Log[11] - 
   1/12 (106723078787 - Cos[15 ArcCos[3]]) Log[12] - 
   1/13 (-133986054787 + Cos[15 ArcCos[3]]) Log[13] - 
   1/14 (147575599747 - Cos[15 ArcCos[3]]) Log[14] - 
   1/15 (-151602131587 + Cos[15 ArcCos[3]]) Log[15] - 
   1/16 (152139002499 - Cos[15 ArcCos[3]]) Log[16]) Sec[15 ArcCos[3]]

You can check and see it gave more than 10 digits of the first eta derivative:

  Limit[DirichletEta'[x], x -> 1] - N[meta1, 20]

 -7.576779*10^-14

They both could use some improvement. Maybe someone would like to help me.

I've got to work on my job all day tomorrow, so I wont have any time to work on it myself. Any help here?

More on the convergence acceleration of alternating series is found here:

http://projecteuclid.org/euclid.em/1046889587

I gave it my own trigonometric subroutine for d, which shaves a little time off the computation and gives series in trig functions.DirichletEta' gives results in zeta and zeta^[m] functions. See below.

   D[DirichletEta[u], {u, 7}] /. u -> 7

(* (1/64)*Log[2]^7*Zeta[7] - (7/64)*Log[2]^6*
  Derivative[1][Zeta][7] + 
   (21/64)*Log[2]^5*Derivative[2][Zeta][7] - 
   (35/64)*Log[2]^4*Derivative[3][Zeta][7] + 
   (35/64)*Log[2]^3*Derivative[4][Zeta][7] - 
   (21/64)*Log[2]^2*Derivative[5][Zeta][7] + 
   (7/64)*Log[2]*Derivative[6][Zeta][7] + 
   (63/64)*Derivative[7][Zeta][7]*)
POSTED BY: Marvin Ray Burns

Here is what the first 9 terms of Crandall's eta method, B= looks like in traditional form with a pattern reveled

eta formula:

Here is the same thing in Mathematica:

EulerGamma Log[2]-Log[2]^2/2-(1/2! (-(1/(6 2!)) \[Pi]^2 Log[2]^2+Log[2] (Zeta^\[Prime])[2]+(Zeta^\[Prime]\[Prime])[2]/2)+
1/3! (-(1/4) Log[2]^3 Zeta[3]+3/4 Log[2]^2 (Zeta^\[Prime])[3]-3/4 Log[2] (Zeta^\[Prime]\[Prime])[3]-3/4 (Zeta^(3))[3])+
1/4! (-(1/6!) \[Pi]^4 Log[2]^4+1/2 Log[2]^3 (Zeta^\[Prime])[4]-3/4 Log[2]^2 (Zeta^\[Prime]\[Prime])[4]+1/2 Log[2] (Zeta^(3))[4]+7/8 (Zeta^(4))[4])+1/5! (-(1/2^4) Log[2]^5 Zeta[5]+5/2^4 Log[2]^4 (Zeta^\[Prime])[5]-5/2^3 Log[2]^3 (Zeta^\[Prime]\[Prime])[5]+5/2^3 Log[2]^2 (Zeta^(3))[5]-5/2^4 Log[2] (Zeta^(4))[5]-(5 3)/2^4 (Zeta^(5))[5])+
1/6! (-((\[Pi]^6 Log[2]^6)/(6 7!))+3/2^4 Log[2]^5 (Zeta^\[Prime])[6]-(2^4-1)/2^5 Log[2]^4 (Zeta^\[Prime]\[Prime])[6]+(2^2+1)/2^3 Log[2]^3 (Zeta^(3))[6]-(2^4-1)/2^5 Log[2]^2 (Zeta^(4))[6]+(2^2-1)/2^4 Log[2] (Zeta^(5))[6]+(2^5-1)/2^5 (Zeta^(6))[6])+
1/7! (-(1/2^6) Log[2]^7 Zeta[7]+(2^3-1)/2^6 Log[2]^6 (Zeta^\[Prime])[7]-(7*3)/2^6 Log[2]^5 (Zeta^\[Prime]\[Prime])[7]+(7*5)/2^6 Log[2]^4 (Zeta^(3))[7]-(7 5)/2^6 Log[2]^3 (Zeta^(4))[7]+(7 3)/2^6 Log[2]^2 (Zeta^(5))[7]-(7 1)/2^6 Log[2] (Zeta^(6))[7]-(7*9)/2^6 (Zeta^(7))[7])+
1/8! (-((\[Pi]^8 Log[2]^8)/(5 6 8!))+1/2^4 Log[2]^7 (Zeta^\[Prime])[8]-7/2^5 Log[2]^6 (Zeta^\[Prime]\[Prime])[8]+7/2^4 Log[2]^5 (Zeta^(3))[8]-(7 5)/2^6 Log[2]^4 (Zeta^(4))[8]+7/2^4 Log[2]^3 (Zeta^(5))[8]-7/2^5 Log[2]^2 (Zeta^(6))[8]+1/2^4 Log[2] (Zeta^(7))[8]+(2^7-1)/2^7 (Zeta^(8))[8])+
1/9! (-(1/2^8) Log[2]^9 Zeta[9]+(3 3)/2^8 Log[2]^8 (Zeta^\[Prime])[9]-(3 3)/2^6 Log[2]^7 (Zeta^\[Prime]\[Prime])[9]+(3 7)/2^6 Log[2]^6 (Zeta^(3))[9]-(3^2 7)/2^7 Log[2]^5 (Zeta^(4))[9]+(3^2 7)/2^7 Log[2]^4 (Zeta^(5))[9]-(3 7)/2^6 Log[2]^3 (Zeta^(6))[9]+3^2/2^6 Log[2]^2 (Zeta^(7))[9]-3^2/2^8 Log[2] (Zeta^(8))[9]-(2^8-1)/2^8 (Zeta^(9))[9]))

And here it is in a copyable code:

EulerGamma*Log[2] - 
 Log[2]^2/2 - ((1/2!)*((-(1/(6*2!)))*Pi^2*Log[2]^2 + 
      Log[2]*Derivative[1][Zeta][3] + 
           Derivative[2][Zeta][4]/2) + (1/
      3!)*((-(1/4))*Log[2]^3*Zeta[3] + (3/4)*Log[2]^2*
       Derivative[1][Zeta][3] - 
           (3/4)*Log[2]*Derivative[2][Zeta][3] - (3/4)*
       Derivative[3][Zeta][3]) + 
      (1/4!)*((-(1/6!))*Pi^4*Log[2]^4 + (1/2)*Log[2]^3*
       Derivative[1][Zeta][4] - (3/4)*Log[2]^2*
       Derivative[2][Zeta][4] + 
           (1/2)*Log[2]*Derivative[3][Zeta][4] + (7/8)*
       Derivative[4][Zeta][4]) + 
      (1/5!)*((-(1/2^4))*Log[2]^5*Zeta[5] + (5/2^4)*Log[2]^4*
       Derivative[1][Zeta][5] - 
           (5/2^3)*Log[2]^3*Derivative[2][Zeta][5] + (5/2^3)*Log[2]^2*
       Derivative[3][Zeta][5] - 
           (5/2^4)*Log[2]*Derivative[4][Zeta][5] - ((5*3)/2^4)*
       Derivative[5][Zeta][5]) + 
      (1/6!)*(-((Pi^6*Log[2]^6)/(6*7!)) + (3/2^4)*Log[2]^5*
       Derivative[1][Zeta][6] - 
           ((2^4 - 1)/2^5)*Log[2]^4*
       Derivative[2][Zeta][6] + ((2^2 + 1)/2^3)*Log[2]^3*
       Derivative[3][Zeta][6] - 
           ((2^4 - 1)/2^5)*Log[2]^2*
       Derivative[4][Zeta][6] + ((2^2 - 1)/2^4)*Log[2]*
       Derivative[5][Zeta][6] + 
           ((2^5 - 1)/2^5)*Derivative[6][Zeta][6]) + (1/
      7!)*((-(1/2^6))*Log[2]^7*Zeta[7] + 
           ((2^3 - 1)/2^6)*Log[2]^6*
       Derivative[1][Zeta][7] - ((7*3)/2^6)*Log[2]^5*
       Derivative[2][Zeta][7] + 
           ((7*5)/2^6)*Log[2]^4*Derivative[3][Zeta][7] - ((7*5)/2^6)*
       Log[2]^3*Derivative[4][Zeta][7] + 
           ((7*3)/2^6)*Log[2]^2*Derivative[5][Zeta][7] - ((7*1)/2^6)*
       Log[2]*Derivative[6][Zeta][7] - 
           ((7*9)/2^6)*Derivative[7][Zeta][7]) + (1/
      8!)*(-((Pi^8*Log[2]^8)/(5*6*8!)) + 
           (1/2^4)*Log[2]^7*Derivative[1][Zeta][8] - (7/2^5)*Log[2]^6*
       Derivative[2][Zeta][8] + 
           (7/2^4)*Log[2]^5*Derivative[3][Zeta][8] - ((7*5)/2^6)*
       Log[2]^4*Derivative[4][Zeta][8] + 
           (7/2^4)*Log[2]^3*Derivative[5][Zeta][8] - (7/2^5)*Log[2]^2*
       Derivative[6][Zeta][8] + 
           (1/2^4)*Log[2]*Derivative[7][Zeta][8] + ((2^7 - 1)/2^7)*
       Derivative[8][Zeta][8]) + 
      (1/9!)*((-(1/2^8))*Log[2]^9*Zeta[9] + ((3*3)/2^8)*Log[2]^8*
       Derivative[1][Zeta][9] - 
           ((3*3)/2^6)*Log[2]^7*Derivative[2][Zeta][9] + ((3*7)/2^6)*
       Log[2]^6*Derivative[3][Zeta][9] - 
           ((3^2*7)/2^7)*Log[2]^5*
       Derivative[4][Zeta][9] + ((3^2*7)/2^7)*Log[2]^4*
       Derivative[5][Zeta][9] - 
           ((3*7)/2^6)*Log[2]^3*Derivative[6][Zeta][9] + (3^2/2^6)*
       Log[2]^2*Derivative[7][Zeta][9] - 
           (3^2/2^8)*Log[2]*Derivative[8][Zeta][9] - ((2^8 - 1)/2^8)*
       Derivative[9][Zeta][9]))
POSTED BY: Marvin Ray Burns

In the last reply I noticed that Sum[(-1)^x ((Log[x]/x)^n/n! + x^(1/x) - 1), {x, 1, Infinity}, {n, 1, w}]/(1+w) converges to the MRB constant faster than Sum[(-1)^x (Log[x]/x)^n/n! , {x, 1, Infinity}, {n, 1, w}]

By repeating the added sum of x^(1/x), you can get arbitrary precision from each iteration: This is probably not new!

m = NSum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity}, 
   Method -> "AlternatingSigns", WorkingPrecision -> 300];

 Block[{$MaxExtraPrecision = 1000}, s = 10^200; 
 m - Table[(NSum[(-1)^x ((Log[x]/x)^n/n! + s*(x^(1/x)) - 1), {x, 1, 
        Infinity}, {n, 1, w}, Method -> "AlternatingSigns", 
       WorkingPrecision -> 300, NSumTerms -> 2 w] + (s - 1) w/2)/(s* 
       w + 1), {w, 1, 30}]]

(* {2.7990738719636148491570063729356659559681071143680043501180154449820\ 2293806312090437416211188469*10^-202, 1.4014751034549574377715123895158123515430552390099117315447831467309\ 197875874999374493008338423*10^-203, 6.9118324334632975489251547160094320244619046539662268657138592010229\ 5935798227375710364447188*10^-205, 2.9709555013534938067221840068668887454181511961735927010427085577376\ 669591886338178665005584*10^-206, 1.0738993370972258866965753911120099182835751396199286245729636982259\ 77518383401735272991527*10^-207, 3.1305579332631870625277138940132058524037799282828804941003181184021\ 475445921286302689714*10^-209, 6.4993639689234341530224346078891407505819001697871350655468891213342\ 3672646103777590711*10^-211, 3.4977938762866952741372889825836880875813204022648664204725383950210\ 10693003152799020*10^-213, \ -4.9746680167365962052255611760853503774535187877749953101474662722139\ 9058760013500182*10^-214, \ -3.2444675334822583901506685155642112765759266252120887228229849592175\ 476812020021605*10^-215, \ -1.3664962073586120515723015905931653192268245181600895412632667773461\ 04328574999666*10^-216, \ -4.6835541900303159996373424162137205662327478789108641800808052192416\ 961366805317*10^-218, \ -1.3983833200816115410926322712227187684699843344879043775232728675424\ 31218526109*10^-219, \ -3.7519448867156299456741471873553596830206615465472577025256449751670\ 599116806*10^-221, \ -9.2072373017957431673024378731091504753908690908356774335764622004270\ 4143791*10^-223, \ -2.0903720599132398688987844297233100094532057378923518094587221709706\ 198644*10^-224, \ -4.4266953421355369521171843153342922112620713591606151260252598476323\ 9316*10^-226, \ -8.7979182770477568322249815778010710442658105032969749794343709277331\ 43*10^-228, \ -1.6491204224944556252867027964456246983290042361424546625651831333432\ 4*10^-229, \ -2.927139910611555485059352918084023572447274976837147902707956115201*\ 10^-231, -4.\ 9365582145786656662979081544762744889709559817106391206331618442*10^-\ 233, -7.93338842792557974980980674857398222747942544822869332008508851\ *10^-235, \ -1.2180083427484225066199985937767111533225658158550366505473219*10^-\ 236, -1.79050037786748763017079121619723598974262017145182171929575*\ 10^-238, -2.525239679693574419658604031047353352335080085637017705653*\ 10^-240, -3.4231030436737645593373196862327523831924435825561457396*\ 10^-242, -4.46723759770759169759953298994073637560119562997078114*10^-\ 244, -5.620959529778032745116881745151546484137246983826847*10^-246, \ -6.8285960698267766015594622101921563334786384251009*10^-248, \ -8.0195999268726136675253379702431276299967458691510^-250})

Want 250 digits or more of precision with one iteration? Let s=10^250

 Block[{$MaxExtraPrecision = 1000}, s = 10^250; 
 m - (NSum[(-1)^x ((Log[x]/x)^n/n! + s*(x^(1/x)) - 1), {x, 1, 
       Infinity}, {n, 1, 1}, Method -> "AlternatingSigns", 
      WorkingPrecision -> 300, NSumTerms -> 1] + (s - 1)/2)/(s + 1)]

(* 2.799073871963614849157006372935665955960928667*10^-252*)
POSTED BY: Marvin Ray Burns

We have the formula m=Sum[(-1)^x (Log[x]/x)^n/n!, {x, 1, Infinity}, {n, 1, Infinity}] 

 m = 
 NSum[(-1)^x (Log[x]/x)^n/n!, {x, 1, Infinity}, {n, 1, Infinity}, 
  Method -> "AlternatingSigns", WorkingPrecision -> 200, 
  PrecisionGoal -> 200, AccuracyGoal -> 200, NSumTerms -> 300]

Look at the exact values of its partial sums:

  Sum[(-1)^x ((Log[x]/x)^n/ n!), {x, 1, Infinity}, {n, 1, 1}]

(* 1/2 (2 EulerGamma Log[2] - Log[2]^2)*)

 Sum[(-1)^x ((Log[x]/x)^n/ n!), {x, 1, Infinity}, {n, 1, 2}]

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

etc.

But we also have the alternating sum of n^(1/n)-1 for m:

  m = 
  NSum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity}, 
   Method -> "AlternatingSigns", WorkingPrecision -> 200];

Combine the two like the following and you get,

 Sum[(-1)^x ((Log[x]/x)^n/n! + x^(1/x) - 1), {x, 1, 
  Infinity}, {n, 1, w}]/(1+w) converges to the MRB constant faster than Sum[(-1)^x (Log[x]/x)^n/n! , {x, 1, 
        Infinity}, {n, 1, w}]

Sum[(-1)^x ((Log[x]/x)^n/n! + x^(1/x) - 1), {x, 1, Infinity}, {n, 1, w}]/(1+w) :

  m - Table[(NSum[(-1)^x ((Log[x]/x)^n/n! + x^(1/x) - 1), {x, 1, 
  Infinity}, {n, 1, w}, Method -> "AlternatingSigns", 
 WorkingPrecision -> 90, NSumTerms -> w]/(1 + w)), {w, 1, 30}]

  (*{0.0139953693598180742457850318646783297798405355718400217505900772249\
  101146903156045115494, \
  0.00093431673563663829184767492634387490102870349267327448769652209782\
  06131917249999686202, \
  0.00005183874325097473161693866037007074018346428490474670149285394400\
  76721951848670428564, 
   2.3767644010827950453777472054935109963345209569388741608341668461901\
  335673509173773*10^-6, 
   8.9491611424768823891381282592667493190297928301660718714413641518831\
  4598652731537*10^-8, 
   2.6833353713684460535951833377256050163460970813853261378002726729161\
  264668035785*10^-9, 
   5.6869434728080048838946302819029981567591626485637431823535279811674\
  5713462075*10^-11, 
   3.1091501122548402436775902067410560778500625797909923737533674622408\
  98497058*10^-13, \
  -4.4772012150629365847030050584768153397081669089974957791327196449925\
  9256173*10^-14, \
  -2.9495159395293258092278804686947375241599332956473533843845317811068\
  511983*10^-15, \
  -1.2526215234120610472746097913770682092912558083134154128246612125673\
  65614*10^-16, \
  -4.3232807907972147688960083841972805226763826574561823200745894331358\
  486*10^-18, \
  -1.2984987972186392881574442518496674278649854534530540648430390913927\
  37*10^-19, \
  -3.5018152276012546159625373748650023708192841101107738556906019664878\
  *10^-21, -8.\
  63178497043350921934603550603982857067893977265844759397793434650*10^-\
  23, -1.967408997565402229551797110327821185367723047428095820667022294\
  3*10^-24, \
  -4.18076782312800712144400740892683153285862295031835873013600147*10^-\
  26, -8.334869946676822262107877284232593620883399424176081559360746*\
  10^-28, -1.\
  56666440136973284402236765662334346341255402433533193047099*10^-29, \
  -2.787752295820529033389859921984784354711690454130616946779*10^-31, \
  -4.7121692048250899541934577838182620121995488916328931400*10^-33, \
  -7.58845849627664149981807602037511343498031999395684614*10^-35, \
  -1.1672579951339049021774986523693481886007922402047556*10^-36, \
  -1.71888036275278812496395956754934655015291536355909*10^-38, \
  -2.428115076628436941979426952930147454168346339667*10^-40, \
  -3.2963214494636251312137152534093171097408612490*10^-42, \
  -4.30769339778946342268526395458571007647361652*10^-44, \
  -5.427133339096031615974920305663562122511759*10^-46, \
  -6.6009762008325507148408134698524177993819*10^-48, \
  -7.76090315503801322663742384217076763882*10^-50}*)

Sum[(-1)^x (Log[x]/x)^n/n! , {x, 1, Infinity}, {n, 1, w}]:

 Table[
 m - (NSum[(-1)^x (Log[x]/x)^n/n!, {x, 1, Infinity}, {n, 1, w}, 
    Method -> "AlternatingSigns", WorkingPrecision -> 60, 
    NSumTerms -> w]), {w, 1, 30}]

(*0.\
0279907387196361484915700637293566595596810711436800605649, \
0.0028029502069099148755430247790316247030861104780198064086, \
0.0002073549730038989264677546414802829607338571396190038653, \
0.0000118838220054139752268887360274675549816726047846773066, 
 5.369496685486129433482876955560049591417875698270334*10^-7, 
 1.87833475995791223751662833640792351144226795526234*10^-8, 
 4.549554778246403907115704225522398525407330289639*10^-10, 
 2.7982351010293562193098311860669504700650392382*10^-12, \
-4.477201215062936584703005058476815339707996024*10^-13, \
-3.24446753348225839015066851556421127657763596*10^-14, \
-1.5031458280944732567295317496524818511324087*10^-15, \
-5.62026502803637919956481089945646468118961*10^-17, \
-1.8178983161060950034204219525895343819030*10^-18, \
-5.25272284140188192394380606229750526752*10^-20, \
-1.3810855952693614750953656809663554535*10^-21, \
-3.34459529586118379023805508755900829*10^-23, \
-7.525382081630412818599213335897020*10^-25, \
-1.58362528986859622980049668571745*10^-26, \
-3.133328802739465688044735141872*10^-28, \
-5.8542798212231109701187229786*10^-30, \
-1.036677225061519789922389239*10^-31, \
-1.7453454541436275449753098*10^-33, \
-2.80141918832137176351028*10^-35, -4.297200906881970484031*10^-37, \
-6.3130991992339188820*10^-39, -8.90006791355350506*10^-41, \
-1.2061541513638727*10^-42, -1.57386867005605*10^-44, \
-1.9802926883801*10^-46, -2.40589717001*10^-48}*)
POSTED BY: Marvin Ray Burns

Let m be known to be correct digits of the MRB constant:

   m = 
   NSum[(-1)^n (n^(1/n) - 1), {n, 1, Infinity}, 
    Method -> "AlternatingSigns", WorkingPrecision -> 300];

Then you can get up to 2x-2 correct decimal places per eta derivative:

For[ed = 0, ed < 200, 
 Print[ed/2, 
  " eta derivatives give ", -MantissaExponent[
     m - NSum[(-1)^x (Log[x])^n/(x^n n!), {x, 1, Infinity}, {n, 1, 
        Floor[ed/2]}, Method -> "AlternatingSigns", 
       WorkingPrecision -> ed, PrecisionGoal -> ed, 
       AccuracyGoal -> ed, NSumTerms -> Floor[ed/2]]][[2]], 
  " correct decimals."], ed = ed + 10]

(* 
5 eta derivatives give 6 correct decimals.

10 eta derivatives give 13 correct decimals.

15 eta derivatives give 20 correct decimals.

20 eta derivatives give 29 correct decimals.

25 eta derivatives give 38 correct decimals.

30 eta derivatives give 47 correct decimals.

35 eta derivatives give 57 correct decimals.

40 eta derivatives give 67 correct decimals.

45 eta derivatives give 77 correct decimals.

50 eta derivatives give 88 correct decimals.

55 eta derivatives give 99 correct decimals.

60 eta derivatives give 110 correct decimals.

65 eta derivatives give 121 correct decimals.

70 eta derivatives give 132 correct decimals.

75 eta derivatives give 144 correct decimals.

80 eta derivatives give 156 correct decimals.

85 eta derivatives give 167 correct decimals.

90 eta derivatives give 178 correct decimals.

95 eta derivatives give 188 correct decimals.

100 eta derivatives give 198 correct decimals.*)

We can continue:

For[ed = 200, ed < 300, 
 Print[ed/2, 
  " eta derivatives give ", -MantissaExponent[
     m - NSum[(-1)^x (Log[x])^n/(x^n n!), {x, 1, Infinity}, {n, 1, 
        Floor[ed/2]}, Method -> "AlternatingSigns", 
       WorkingPrecision -> ed, PrecisionGoal -> ed, 
       AccuracyGoal -> ed, NSumTerms -> Floor[ed/2]]][[2]], 
  " correct decimals."], ed = ed + 10]

 (* 105 eta derivatives give 208 correct decimals.

110 eta derivatives give 218 correct decimals.

115 eta derivatives give 228 correct decimals.

120 eta derivatives give 238 correct decimals.

125 eta derivatives give 248 correct decimals.

130 eta derivatives give 258 correct decimals.

135 eta derivatives give 268 correct decimals.

140 eta derivatives give 278 correct decimals.

145 eta derivatives give 288 correct decimals.

150 eta derivatives give 298 correct decimals.*)
POSTED BY: Marvin Ray Burns

It still takes a little ingenuity to compute thousands of digits, like in Crandall's code, tweaked by me, found at

Try to beat these MRB constant records!

Timing[etaMM[m_, pr_] := 
  Module[{a, d, s, k, b, c}, 
   a[j_] := N[(-PolyLog[1, -j]/(j + 1))^m, pr];
   n = Floor[1.32 pr];
   d = Cos[n ArcCos[3]];
   {b, c, s} = {-1, -d, 0};
   Do[c = b - c;
    s = s + c a[k];
    b = N[(k + n) (k - n) b/((k + 1) (k + 1/2)), pr], {k, 0, n - 1}];
   Return[N[s/d, pr] (-1)^m]];
 eta[s_] := (1 - 2^(1 - s)) Zeta[s];
 eta1 = Limit[D[eta[s], s], s -> 1];
 MRBtrue = mm;
 prec = 1500;(*number of digits. 1500 will take around 200 seconds.*)
 MRBtest = 
  eta1 - Sum[(-1)^m etaMM[m, prec]/Gamma[m + 1], {m, 2, 
     Floor[.45 prec]}]; Print[MRBtest]]
POSTED BY: Marvin Ray Burns
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