I was having trouble formatting the following in the following thread. I had some text placed immediately after an image.
Using formula 5 on page 3 of http://arxiv.org/pdf/0912.3844v3.pdf
.
We can compute a great deal of digits of the integral analog of the MRB constant' (I once called it the MKB constant, named after Marsha Kell-Burns my, now ex, wife.) In the paper Mathar simply calls it M1.
This time we will compute the Imaginary part first to at least 500 digits:
a[1] = 0; For[n = 1, n < 11,
a[n] = N[2/Pi -
1/Pi*NIntegrate[
Cos[Pix]x^(1/x)(1 - Log[x])/x^2, {x, 1, Infinity},
WorkingPrecision -> 100n], 50 n]; Print[a[n] - a[n - 1]],
n++]; Print[a[11]]
giving
0.6840003894379321291827444599926611267109914826549994343226303771381530581249766381509598342127214787
0.*10^-101
0.*10^-151
0.*10^-201
0.*10^-251
0.*10^-301
0.*10^-351
0.*10^-401
0.*10^-451
0.*10^-501
0.6840003894379321291827444599926611267109914826549994343226303771381530581249766381509598342127214786722379645160914886099586780498831455794087390519118879988351918366211827085883779918191195794251385436100844782462528597869421390620796113023053439642582325892202911183326091512210367124716901047132601108752764946385830438156754378694878046808312868541961166205744280461776232345922905313658259576212809654022016030244583148587352474339130505540080799774619683572540292971258866450201101870835703060314349396491402064932644813564545345219868887520120
.
Likewise the real part:
b[1] = 0; For[n = 1, n < 11,
b[n] = N[-1/Pi*
NIntegrate[Sin[Pi*x]*x^(1/x)*(1 - Log[x])/x^2, {x, 1, Infinity},
WorkingPrecision -> 100*n], 50 n]; Print[b[n] - b[n - 1]],
n++]; Print[b[11]]
giving
0.07077603931152880353952802183028200136575469620336302758317278816361845726438203658083188126617723821
0.*10^-102
0.*10^-152
0.*10^-202
0.*10^-252
0.*10^-302
0.*10^-352
0.*10^-402
0.*10^-452
0.*10^-502
0.07077603931152880353952802183028200136575469620336302758317278816361845726438203658083188126617723820944073396910971792699904464538475364292258443860652193330471222906120205483985764336623434898438270710499897053952312269178485299032185072743545220051257328105422174249313177670295863771714489658779291185716175115405623656039914848817528200250723061535734571065031458992196831648681239079549382556509741967588147362548743205919028695774572411439927516593391029992733107982746794845130889328251307263102570083031527430861023428334369104098217022622689
.
Then the magnitude:
N[Sqrt[a[11]^2 + b[11]^2], 500]
giving
0.68765236892769436980931240936544016493963738490362254179507101010743\
3662534784937068627298240498468188731929334335466123286287665409457565\
9577211580255650416284625143925097120589697986500952590195706813170472\
5387265069668971286335322245474865156721299946377659227025219748069576\
0895993932096027520027641920489863095279507385793449828250341732295653\
3809181101532087948181335825805498812728097520936901677028741356923292\
2644964771090329726483682930417491673753430878118054062296678424687465\
624513174205
.
That checks with the 200 digits computed by the quadosc command in mpmath by FelisPhasma at https://github.com/FelisPhasma/MKB-Constant .
I just now finished 750 digits, (about the max with formula 5 from the paper, as far as Mathematica is concerned).
Here is the work:
a[1] = 0; For[n = 1, n < 16,
a[n] = N[2/Pi -
1/Pi*NIntegrate[
Cos[Pi*x]*x^(1/x)*(1 - Log[x])/x^2, {x, 1, Infinity},
WorkingPrecision -> 100*n], 50 n]; Print[a[n] - a[n - 1]],
n++]; Print[a[16]];
b[1] = 0; For[n = 1, n < 16,
b[n] = N[-1/Pi*
NIntegrate[Sin[Pi*x]*x^(1/x)*(1 - Log[x])/x^2, {x, 1, Infinity},
WorkingPrecision -> 100*n], 50 n]; Print[b[n] - b[n - 1]],
n++]; Print[b[16]]; Print[N[Sqrt[a[16]^2 + b[16]^2], 750]]
0.6840003894379321291827444599926611267109914826549994343226303771381530581249766381509598342127214787
0.*10^-101
0.*10^-151
0.*10^-201
0.*10^-251
0.*10^-301
0.*10^-351
0.*10^-401
0.*10^-451
0.*10^-501
0.*10^-551
0.*10^-601
3.*10^-650
-4.*10^-700
-2.6*10^-749
0.68400038943793212918274445999266112671099148265499943432263037713815\
3058124976638150959834212721478672237964516091488609958678049883145579\
4087390519118879988351918366211827085883779918191195794251385436100844\
7824625285978694213906207961130230534396425823258922029111833260915122\
1036712471690104713260110875276494638583043815675437869487804680831286\
8541961166205744280461776232345922905313658259576212809654022016030244\
5831485873524743391305055400807997746196835725402929712588664502011018\
7083570306031434939649140206493264481356454534521986888752011950353818\
1776359577265099302389566135475579468144849763261779452665955246258699\
8679271659049208654746533234375478909962633090080006358213908728990850\
5026759549928935029206442637425786005036048098598304092996753145589012\
64547453361707037686708654522699
0.07077603931152880353952802183028200136575469620336302758317278816361845726438203658083188126617723821
0.*10^-102
0.*10^-152
0.*10^-202
0.*10^-252
0.*10^-302
0.*10^-352
0.*10^-402
0.*10^-452
0.*10^-502
2.*10^-551
-1.*10^-600
1.8*10^-650
1.27*10^-699
4.34*10^-749
0.07077603931152880353952802183028200136575469620336302758317278816361\
8457264382036580831881266177238209440733969109717926999044645384753642\
9225844386065219333047122290612020548398576433662343489843827071049989\
7053952312269178485299032185072743545220051257328105422174249313177670\
2958637717144896587792911857161751154056236560399148488175282002507230\
6153573457106503145899219683164868123907954938255650974196758814736254\
8743205919028695774572411439927516593391029992733107982746794845130889\
3282513072631025700830315274308610234283343691040982170226226904594029\
7055093272952022662549075225941956559080574835998923469310063614655255\
0629713179601483134045038416878054929072981851045829413286377842843667\
5378730394247519728064887287780998671021887797977772522419765594172569\
277490031071938177749184834961300
0.687652368927694369809312409365440164939637384903622541795071010107433662534784937068627298240498468188731929334335466123286287665409457565957721158025565041628462514392509712058969798650095259019570681317047253872650696689712863353222454748651567212999463776592270252197480695760895993932096027520027641920489863095279507385793449828250341732295653380918110153208794818133582580549881272809752093690167702874135692329226449647710903297264836829304174916737534308781180540622966784246874656245131742049004832216427665542900559350289936114782223424261285828326467186036500189315374147638489679365569122714398706519530651330568884655048857998738535162606116788633540389660052822237449082894798620397228331715198160243676576563833057235963591510865254600
Using formula 7 from the same paper,
.
(Treating it as we did formula 3),
First, the imaginary part to at least 1000 digits::
a[1] = 0; For[n = 1, n < 21,
a[n] = N[2/Pi +
1/Pi^2 NIntegrate[
Sin[x Pi] x^(1/x) (1 - 3 x + 2 (x - 1) Log[x] + Log[x]^2)/
x^4, {x, 1, Infinity}, WorkingPrecision -> 100 n], 50 n];
Print[a[n] - a[n - 1]], n++]; Print[a[21]]
0.6840003894379321291827444599926611267109914826549994343226303771381530581249766381509598342127214787
0.*10^-101
0.*10^-151
0.*10^-201
0.*10^-251
0.*10^-301
0.*10^-351
0.*10^-401
0.*10^-451
0.*10^-501
0.*10^-551
0.*10^-601
0.*10^-651
0.*10^-701
0.*10^-751
0.*10^-801
0.*10^-851
0.*10^-901
-2.*10^-950
5.*10^-1000
0.684000389437932129182744459992661126710991482654999434322630377138153058124976638150959834212721478672237964516091488609958678049883145579408739051911887998835191836621182708588377991819119579425138543610084478246252859786942139062079611302305343964258232589220291118332609151221036712471690104713260110875276494638583043815675437869487804680831286854196116620574428046177623234592290531365825957621280965402201603024458314858735247433913050554008079977461968357254029297125886645020110187083570306031434939649140206493264481356454534521986888752011950353818177635957726509930238956613547557946814484976326177945266595524625869986792716590492086547465332343754789099626330900800063582139087289908505026759549928935029206442637425786005036048098598304092996753145589012645474533617070376867086545228223060940434935219252885333298390272342234952870883304116640409421452765284609364941205344122569781634782508368641126766528707019957340895061936246645065753101916781254557006989818409283317145837167345971516970849116096077030635788389165381066055992688
Then the real part to at least 1000 digits:
b[1] = 0; For[n = 1, n < 21,
b[n] = N[1/Pi^2 -
1/Pi^2 NIntegrate[
Cos[Pi x] x^(1/x) (1 - 3 x + 2 (x - 1) Log[x] + Log[x]^2)/
x^4, {x, 1, Infinity}, WorkingPrecision -> 100 n], 50 n];
Print[b[n] - b[n - 1]], n++]; Print[b[21]]
0.07077603931152880353952802183028200136575469620336302758317278816361845726438203658083188126617723821
0.*10^-102
0.*10^-152
0.*10^-202
0.*10^-252
0.*10^-302
0.*10^-352
0.*10^-402
0.*10^-452
0.*10^-502
0.*10^-552
0.*10^-602
0.*10^-652
0.*10^-702
0.*10^-752
0.*10^-802
0.*10^-852
-3.*10^-901
8.*10^-951
-4.6*10^-1000
0.0707760393115288035395280218302820013657546962033630275831727881636184572643820365808318812661772382094407339691097179269990446453847536429225844386065219333047122290612020548398576433662343489843827071049989705395231226917848529903218507274354522005125732810542217424931317767029586377171448965877929118571617511540562365603991484881752820025072306153573457106503145899219683164868123907954938255650974196758814736254874320591902869577457241143992751659339102999273310798274679484513088932825130726310257008303152743086102342833436910409821702262269045940297055093272952022662549075225941956559080574835998923469310063614655255062971317960148313404503841687805492907298185104582941328637784284366753787303942475197280648872877809986710218877979777725224197655941725692774900310719381777491848349627938468198411955193898347075098152638657614980900350262780319142430252921925131515239611841070722530473939496294305264627977744876814858325335947117076721493110160508928494597906728688873533031986215124467678736429981544321187124269147141804397293341613
Then the magnitude:
In[97]:= N[Sqrt[a[21]^2 + b[21]^2], 1000]
Out[97]= 0.\
6876523689276943698093124093654401649396373849036225417950710101074336\
6253478493706862729824049846818873192933433546612328628766540945756595\
7721158025565041628462514392509712058969798650095259019570681317047253\
8726506966897128633532224547486515672129994637765922702521974806957608\
9599393209602752002764192048986309527950738579344982825034173229565338\
0918110153208794818133582580549881272809752093690167702874135692329226\
4496477109032972648368293041749167375343087811805406229667842468746562\
4513174204900483221642766554290055935028993611478222342426128582832646\
7186036500189315374147638489679365569122714398706519530651330568884655\
0488579987385351626061167886335403896600528222374490828947986203972283\
3171519816024367657656383305723596359151086525460036387486837632622334\
2987257095524637683005910353149353985736118868884201748241906260834981\
7303422370398413326428269921074045506558966667483453656748906071577744\
4147548424388220133662816274116986724576330176058912438027319979840883\
05950589130911719199