I was also having issues with the following post. I had to remove some data before it appeared nice again.
Where the upper limit of the following integrals shows Infinity, it is meant to be the (Ultraviolet limit of the sequence)
as mentioned by Mathar here:
![enter image description here](/c/portal/getImageAttachment?filename=29867.JPG&userId=366611)
I derived a new formula for computing the integral analog of the MRB constant':
f[x_]:=x^(1/x);-((2 I)/\[Pi]^3) + 1/\[Pi]^2 - (
2 I)/\[Pi] + (I/Pi)^3*
Integrate[(-1)^x*D[f[x], {x, 3}], {x, 1, Infinity}]
In traditional form that is M1=
![enter image description here](/c/portal/getImageAttachment?filename=84695.JPG&userId=366611)
Using it I computed 2000 digits in only 10.8 minutes:
In[131]:= Timing[f[x_] = x^(1/x);
a = N[1/\[Pi]^2 + (1/Pi)^3*
NIntegrate[Sin[Pi*x]*D[f[x], {x, 3}], {x, 1, Infinity},
WorkingPrecision -> 4000], 2000];
b = N[2/\[Pi]^3 +
2/\[Pi] + (1/Pi)^3*
NIntegrate[Cos[Pi x]*D[f[x], {x, 3}], {x, 1, Infinity},
WorkingPrecision -> 4000], 2000];
Print[N[Sqrt[a^2 + b^2], 2000]]]
During evaluation of In[131]:= 0.68765236892769436980931240936544016493963738490362254179507101010743366253478493706862729824049846818873192933433546612328628766540945756595772115802556504162846251439250971205896979865009525901957068131704725387265069668971286335322245474865156721299946377659227025219748069576089599393209602752002764192048986309527950738579344982825034173229565338091811015320879481813358258054988127280975209369016770287413569232922644964771090329726483682930417491673753430878118054062296678424687465624513174204900483221642766554290055935028993611478222342426128582832646718603650018931537414763848967936556912271439870651953065133056888465504885799873853516260611678863354038966005282223744908289479862039722833171519816024367657656383305723596359151086525460036387486837632622334298725709552463768300591035314935398573611886888420174824190626083498173034223703984133264282699210740455065589666674834536567489060715777444147548424388220133662816274116986724576330176058912438027319979840883059505891309117191987761469414772648989343657425085034050732738529903546587114217499635584514475429656959327732862489935076490012861232249244670423220090484477969004477448946670434279197103332581857937517719898657425832767700119265854957115794801143278185461993723493131802360791389248808154759564302727311223193005229640892474022665093207969297797972308795483218256171403916521459251943207234100609086755844459050004667079633465456383179509789357941736916352744611848521664077918386624294040883487647062354653558109265769644276994369741555722263494599492834558291937955573706480722982389806312472239746286527176248883116124285469947303667188075506826507811479428582807366599407544908560990699866167233307144245764835741501174979679166078765231145175411199825822532170091858833628202128777966026600647843068442894310401343003939117236867245656732686719139206716028255819141802331701942027248337771633882445225049334329008827371320849006472846226868011129149192754883153995560921671208059671732704499253517327447921147157
Out[131]= {653.145, Null}
I am presently computing 10,000 digits using that formula. Come back here for results!
That formula didn't work out; I will try one of the following formulas.
Here are 2 more, more advanced formulas; remember f(x) is x^(1/x):
![enter image description here](/c/portal/getImageAttachment?filename=40936.JPG&userId=366611)
I did finish a 5,000 digit computation using M1=
![enter image description here](/c/portal/getImageAttachment?filename=61048.JPG&userId=366611)
in 48.11 minutes.
Here are the 5000 digits:of the magnitude:
0.68765236892769436980931240936544016493963738490362254179507101010743366253478493706862729824049846818873192933433546612328628766540945756595772115802556504162846251439250971205896979865009525901957068131704725387265069668971286335322245474865156721299946377659227025219748069576089599393209602752002764192048986309527950738579344982825034173229565338091811015320879481813358258054988127280975209369016770287413569232922644964771090329726483682930417491673753430878118054062296678424687465624513174204900483221642766554290055935028993611478222342426128582832646718603650018931537414763848967936556912271439870651953065133056888465504885799873853516260611678863354038966005282223744908289479862039722833171519816024367657656383305723596359151086525460036387486837632622334298725709552463768300591035314935398573611886888420174824190626083498173034223703984133264282699210740455065589666674834536567489060715777444147548424388220133662816274116986724576330176058912438027319979840883059505891309117191987761469414772648989343657425085034050732738529903546587114217499635584514475429656959327732862489935076490012861232249244670423220090484477969004477448946670434279197103332581857937517719898657425832767700119265854957115794801143278185461993723493131802360791389248808154759564302727311223193005229640892474022665093207969297797972308795483218256171403916521459251943207234100609086755844459050004667079633465456383179509789357941736916352744611848521664077918386624294040883487647062354653558109265769644276994369741555722263494599492834558291937955573706480722982389806312472239746286527176248883116124285469947303667188075506826507811479428582807366599407544908560990699866167233307144245764835741501174979679166078765231145175411199825822532170091858833628202128777966026600647843068442894310401343003939117236867245656732686719139206716028255819141802331701942027248337771633882445225049334329008827371320849006472846226868011129149192754883153995560921671208059671732704499253517327447529208297180672654123457301218758892278525894167935930983363218877512533994251978272092700003994136520699813263053327399132641690231179063314931546906927612775633995348209911166678724589467821767106592498663827057034363632241807121831546175498178011687284590439293322231263406301066863589072717290630291441982684113819198880100231182613587798104863611185433976009254862585527222843445901958943153561148829083242874018226480554274231391324767376148485531787767908124831873688579979114662856184612164534836370699371440464263768724668291617743681719766849740663590277737977490693183461320266666793472116774276618408124767965369796362732668987556797338128876129264558867657737417548617146808592137056879602982206609613881069490166381528825180204703315896719667069923077454352649723496033985893188309150391579573916059639453655188856334980355047281560296288150836680499821806918067869468571687709518088408966653716009356556714281694904914038988996962213833530636987279769672200413448893419914190954063100962251649102614676944333201213024711868954772741991675045198246947499574872027800654821823797116399297131866662866832215332914761325880983081211272181775518951539503852063119472301382766303820851467743266039356123495461914463960644386394228342211998370152351720235034997434035743513051754761571835043769475528640144621307760159481496713401409374957729200400650100318226988524015127382509490642900236553851499823658269458873976032051355393161653806016080446394196719312454167915154602448638624354575153334932298393406734174580316934939632892851077461038399470015366439910136971186909599331204517462262508377673477745789645309425145559198802530351403897927622891172233239135167420567162398873965477371498335087310395422796362380227536212159184529243644094285328763286873653399867593200891823468738537356817916009007206857590792983184556882143118383332812491747733056313117179696094921120670802012310012864110800437831852620698327457619035904268498030693438632685623213366864129523404256345542376567721287706234359125016588483777876970236084456277023948551334490591022594253744077631232660869593809453087749830900393202787736482133628148979992109544954840067942735030391105496026321872468122542495017023785810605820545392820104069279893067324597299043883381251767370331206913429284614563732308018369972360638019778425246546329838131639355043236388708044857300408692365733932897876809202025693305332974091411983635619038514442263783801745983300121464879550146672827072002317686396598587702487509572349422593441184802476344187280014450860069307120621758277552124841158659386176036703247124389223327008210072318671884895179305778728051888524412158486781863155034447221379906386062559915129172725833420555901857729690605950941678587057025641848365090809750870051863842805803189784976076099574956436664131457150096711473033060684065060747340764998195621425524824611657787212347497307297184843276100338110267863618974154272345482369968216663233417338501929114697679974461999040589290327155974468087040862022522065912789
I'm getting closer to 10K digits of M1: Using
, where f(x)=x^(1/x).
I got approx. 10K digits of the imaginary part, but the real part was a little garbled.