Revisting my accelerated slog solution using Abel matrix inversion

[sheldonison]

The results match your post at the beginning of this thread for about 25-26 decimal digits; which is interesting and way too good to be due to chance, but the differences will require more investigation... edit: perhaps something like this explains what's going on where theta is a 1-cyclic function.
\( \text{JaySlogSeries}(z)\approx\text{KneserSlog}(z)+\theta(\text{KneserSlog}(z))\;\;\;??? \)
Such a theta mapping might explain why Jay's computation is internally consistent to 450 decimal digits, of which Jay posted ~78 decimal digits here on the forum.  Perhaps one could calculate such a theta mapping to 

The \( \text{KneserSlog(z)+\theta(\text{KneserSlog(z)) \) equation looks like it is a very good approximation of Jay's slog, but the amplitude of the primary terms at O(10^-27) seems larger than one would have expected.  The \( \theta \)=jaytheta fourier series below is accurate to 10^-76 near the real axis, and within 2*10^-69 inside a unit circle around z=0.  It has 17 terms from exp(-8*Pi*I*z)..exp(8*Pi*I*z), where the terms come in complex conjugate pairs.

Are there any other experiments we could try to figure out what's going on in Jay's slog matrix, and why it is coming up with a nearly 1-cyclic mapping of Kneser's slog?

The other thing is that I probably need to explain how my fatou.gp slog uses the Schroeder function along with a different 1-cyclic \( \theta(z) \) mapping to guarantee convergence with Kneser's slog, but I don't want to go off topic.

Code:

{ jaytheta(z) =
( 3.5728276456649148477566116177388023019007268196783349159893701815277141067225 E-95  
 -1.3649661115074318758359394184069459157348183079287255024123555076990915418413 E-95*I)*exp(z*2*Pi*I*-8) +
(-1.5440810246150882539482350614654446227922901045111993369886391972973287458105 E-86  
 -5.5669023559216323925345216438025695019598405830531829949210261661331938210694 E-87*I)*exp(z*2*Pi*I*-7) +
( 3.7086748176968958397009105825753164147304702764047775074802376855971837334383 E-78  
 +1.1451582025980524236616989591346313817219047309050558224544165806126816293184 E-77*I)*exp(z*2*Pi*I*-6) +
( 1.1210614871772847624343387516545807440942656652432663519744124767146307448107 E-68  
 -1.1470591130474478937181194872389544976665521110710100672287841946582901951881 E-68*I)*exp(z*2*Pi*I*-5) +
(-3.8539202891193833420560824550695624902356998189778776872572042272613662988617 E-59  
 -2.1138152531049279432069204778324824829587238753274203674250488900370678846025 E-59*I)*exp(z*2*Pi*I*-4) +
(-1.4211380450705848101841053395855311988186232917200164673339875435393048716064 E-49  
 +2.6322911718861857981740665561538780260969980190539409008516880732957221621375 E-49*I)*exp(z*2*Pi*I*-3) +
( 4.6199048930995315941719898370835132530766708685634714050488164704159381114052 E-39  
 +5.2167577270913630361688519902263191672290188822400996799706708147241652355711 E-39*I)*exp(z*2*Pi*I*-2) +
( 1.0696559455881259713655169257010732622925108009677598645744429747235112837897 E-27  
 +1.9109391105341231522214949744300521979800047513277681914525934159682392769201 E-28*I)*exp(z*2*Pi*I*-1) +
 -2.1393118911854917525169486869813993088913918409366654477997123124873055208004 E-27  +
( 1.0696559455881259713655169257010732622925108009677598647891077428436746514155 E-27  
 -1.9109391105341231522214949744300521979800047513277681928987942986883883045070 E-28*I)*exp(z*2*Pi*I) +
( 4.6199048930995315941719898370835132530766708700408536630674514450527650792040 E-39  
 -5.2167577270913630361688519902263191672290188809445724819576941574423230216014 E-39*I)*exp(z*2*Pi*I*2) +
(-1.4211380450705848101841053395855311988566716857215934631581381619210748914954 E-49  
 -2.6322911718861857981740665561538780261149441871658727620111922865086449876717 E-49*I)*exp(z*2*Pi*I*3) +
(-3.8539202891193833420560824550691649153570853618694845312416706112027037826147 E-59  
 +2.1138152531049279432069204778267325506190281853467293590631743118540833037669 E-59*I)*exp(z*2*Pi*I*4) +
( 1.1210614871772847624343805343719207388438541669933487597084797186218859963216 E-68  
 +1.1470591130474478937181088034863740700773115622928860057550724684519494199295 E-68*I)*exp(z*2*Pi*I*5) +
( 3.7086748176968966527644112254786062183031494760821947404652325628506661872837 E-78  
 -1.1451582025980529478042915261001026578076002314767359370661049975563219165253 E-77*I)*exp(z*2*Pi*I*6) +
(-1.5440810162317820257286076540777670544911171119974240869791763102058717224083 E-86  
 +5.5669023741799632403812586867225091698768319497226142393990713274070953349022 E-87*I)*exp(z*2*Pi*I*7) +
( 3.4996277197133582104849492242489945468307342754067132866754747304481619780121 E-95  
 +1.3265185330275725639281119595111317055761552257026709292842321829759362977176 E-95*I)*exp(z*2*Pi*I*8);
}