Crandall's eta formula finally proven!

[bmmmburns (Marvin Ray Burns A.G.S. (cum laude))]

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

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

@

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

[bmmmburns (Marvin Ray Burns A.G.S. (cum laude))]

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]]

[bmmmburns (Marvin Ray Burns A.G.S. (cum laude))]

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.*)

[bmmmburns (Marvin Ray Burns A.G.S. (cum laude))]

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}*)

[bmmmburns (Marvin Ray Burns A.G.S. (cum laude))]

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
229380631209043741621118846910^-202,
1.4014751034549574377715123895158123515430552390099117315447831467309
19787587499937449300833842310^-203,
6.9118324334632975489251547160094320244619046539662268657138592010229
593579822737571036444718810^-205,
2.9709555013534938067221840068668887454181511961735927010427085577376
66959188633817866500558410^-206,
1.0738993370972258866965753911120099182835751396199286245729636982259
7751838340173527299152710^-207,
3.1305579332631870625277138940132058524037799282828804941003181184021
47544592128630268971410^-209,
6.4993639689234341530224346078891407505819001697871350655468891213342
367264610377759071110^-211,
3.4977938762866952741372889825836880875813204022648664204725383950210
1069300315279902010^-213,
-4.9746680167365962052255611760853503774535187877749953101474662722139
905876001350018210^-214,
-3.2444675334822583901506685155642112765759266252120887228229849592175
47681202002160510^-215,
-1.3664962073586120515723015905931653192268245181600895412632667773461
0432857499966610^-216,
-4.6835541900303159996373424162137205662327478789108641800808052192416
96136680531710^-218,
-1.3983833200816115410926322712227187684699843344879043775232728675424
3121852610910^-219,
-3.7519448867156299456741471873553596830206615465472577025256449751670
59911680610^-221,
-9.2072373017957431673024378731091504753908690908356774335764622004270
414379110^-223,
-2.0903720599132398688987844297233100094532057378923518094587221709706
19864410^-224,
-4.4266953421355369521171843153342922112620713591606151260252598476323
931610^-226,
-8.7979182770477568322249815778010710442658105032969749794343709277331
4310^-228,
-1.6491204224944556252867027964456246983290042361424546625651831333432
410^-229,
-2.927139910611555485059352918084023572447274976837147902707956115201
10^-231, -4.
936558214578665666297908154476274488970955981710639120633161844210^-
233, -7.93338842792557974980980674857398222747942544822869332008508851
10^-235,
-1.218008342748422506619998593776711153322565815855036650547321910^-
236, -1.79050037786748763017079121619723598974262017145182171929575
10^-238, -2.525239679693574419658604031047353352335080085637017705653*
10^-240, -3.4231030436737645593373196862327523831924435825561457396*
10^-242, -4.4672375977075916975995329899407363756011956299707811410^-
244, -5.62095952977803274511688174515154648413724698382684710^-246,
-6.828596069826776601559462210192156333478638425100910^-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*)

[bmmmburns (Marvin Ray Burns A.G.S. (cum laude))]

Here is what the first 9 terms of Crandall’s eta method,

looks like in traditional form with a pattern reveled

:

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]))

[bmmmburns (Marvin Ray Burns A.G.S. (cum laude))]

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

,
where B is the MRB constant and eta^[1] 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:

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^[2] 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]*)

---

1. m ↩︎

2. m ↩︎

[bmmmburns (Marvin Ray Burns A.G.S. (cum laude))]

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

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*)

[bmmmburns (Marvin Ray Burns A.G.S. (cum laude))]

The second formula has been proved, but Crandall didn’t show it, or how the terms simplify, to me.

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,

Try to beat these MRB constant records!

POSTED BY: Marvin Ray Burns, and distinguished colleagues My exciting experiences using Wolfram technologies! Unless otherwise stated, all infographics in this discussion were drafted with the help of AI. Any given notebook in this discussion may have been written by the Wolfram Notebook Assistant. The author of this discussion understands them both and takes full responsibility for all claims made in them. CMRB [Screenshot2024-11-25023042.png] is the MRB constant. A General Decim…

.
See the end, where I began to ask for some help.
.