Identities inspired by Ramanujan Notebooks (part 2)

by Simon Plouffe

April 2006

These are new findings related to the ones found in 1998. I had the idea that maybe the sums in the first note I wrote were looking in the bad direction. They are good yes but maybe there is better.  Actually there is, instead of considering powers of exp(2π) I should consider powers of exp(kπ).

After a few experiments with Maple (computer algebra system) and PSLQ (integer relations algorithm) here are the findings and remarks.

- I could go up to Zeta(39) and powers of exp(30π) after that the numbers are too close together for any computation,  1/exp(30π) is of the order of 10-40 and even with 2000 decimal digits I get an error.

- The powers of exp(kπ)±1 are related to the divisors of a number.

- The numbers that appear to be expressible are so far : log(A) where A is an integer or simple alebraic like arccosh(1/2) or arctanh(2/3), ζ(2n+1), π2n+1, ζ(2n+1) and some formulas involving lnΓ(1/4) and ln(π) but not separately.

Formulas for odd powers of π and ζ(2n+1)

That one is the fastest converging series so far with 5.45 digits for each n.

Other formulas

Here is a table of results with the coefficients :

Results with Zeta(z), Pi^z, S(z,1), S(z,2), S(z,4)
where S(k,m) is

infinity
-----
\               1
)     --------------------   = S(k,m).
/       k
-----   n  (exp(m Pi n) - 1)
n = 1

The display is the integer relation that
make the vector [Zeta(z), Pi^z, S(z,1), S(z,2), S(z,4)] equal to 0.

3, [1, 0, -28, 37, -7]
5, [-10, 0, 240, -259, -1]
7, [-52, 0, 1216, -1339, 19]
9, [27930, -1, 39680, 16401, -221]
11, [-538116, 2, -1108992, 33843, -1083]
13, [-4616430, 2, -27095040, 17859891, 2289]
15, [52761192, -2, 100614144, 4902099, 6141]
17, [-2381720400, 8, 2954035200, -7717548915, 72915]
19, [39005722340, -14, -1613496320, 79624947155, -6155]
21, [-221481750994700, 6732, 850750073733120, -1293713997563111, 421840591]
23, [1226556695374, 9, -84963147907072, 87416281554613, -20256793]
25, [5203911059453925, -1921, -1567010009907200, 11974832749256745, -620441695]
27, [8148451902123610, -306, -1701430826106880, 17998334655707395, -25353295]
29, [-13964212268709884725, -140282, 1170802376844509184000, -1198730801485834158745, 103905205295]
31, [-54171522182777606620, 20123, 56675100822926786560, -165018145241264800615, 52782800815]
33, [-58909621176539933053170, 2545893, -131688610996976101621760, 13869368616464330113914, 27431905401506]

Results with Zeta(z), S(z,1), S(z,2), S(z,4).
The display is the integer relation that
make the vector [Zeta(z), S(z,1), S(z,2), S(z,4)] equal to 0.

3, [1, -28, 37, -7]
5, [-10, 240, -259, -1]
7, [-52, 1216, -1339, 19]
9, [34440, -800000, 869253, -373]
11, [-64144, 1487872, -1617613, 1453]
13, [-4952480, 114831360, -124738499, 2179]
15, [-9672256, 224247808, -243606007, 13687]
17, [-95521188480, 2214570557440, -2405615921073, 2986673]
19, [87161441536, -2020746723328, 2195077314937, -7708537]
21, [-511417392598515200, 11856658470320209920, -12879494224969023401, 969451783081]
23, [-809054768954368, 18757052482453504, -20375166492392041, 4472029801]
25, [-9402626177680619520, 217989612878618951680, -236794866356583343203, 1122603152483]
27, [-433559729330180096, 10051608326929645568, -10918727935370641697, 149780635937]
29, [70591765772239009938104320, -1636592903864970323479756800, 1777776435935195869731249349, -525747526375283909]
31, [-520167571795410350915584, 12059516381224761076744192, -13099851536046881623355143, 11231299844779783]
33, [-10652244887519367873408902922240, 246960650058691978789414796001280, -268265139838691529135407355836517, 4960814599174753990757]
35, [-2732942456837303169145438208, 63360282530610722654381932544, -68826167447973382833596090501, 3688053840923281541]
37, [-1424192973610401074580480646493634560, 33018356811066418402946396259121889280, -35866742758328669027445599763623152661, 41448475338242211513994261]
39, [31536188043819957788458947444736, -731132036585425088557545661923328, 794204412675724846988747136207187, -2659842854283579394387]
41, [12773377697620891851198465395561679146188800, -296136794882949433788107046141906375032176640, 321683550278214452268691394539562438746427097, -23234778187417606532705421872857]
43, [233097371723978108279933298059247616, -5404107683343089621820512227764994048, 5870302426792274590206831552234790117, -1228751826452728351300837]
45, [140600817767916545581693183361009315526861326909440, -3259676220131698228640954778884807655906629501583360, 3540877855667547304242700329643553467180817656754611, -15984438359184036727180220465501352371]
47, [2254919587126878277155357568098096977543168, -52277844988066489224430209234308641738194944, 56787684162320988691600873634610990609888551, -742912859949264106154916607271]
49, [284524049801951474153643015128700095967174408724807680, -6596378982132770562246518166744835627859463792649830400, 7165427081736675532219530224305149732457982689130518611, -2021665726027302913912664170079031072851]
51, [12213435551464851306588720257503511850254336, -283155148492348076155312278372927288186503168, 307582019595278030260782036775946315366307047, -251492292317888012003479295207]
53, [130137454758409843860165531701040966161594385248587486330880, -3017094589909565702616435024906390294964042473906645824962560, 3277369499426385448129400121915923378120436526114126848209189, -57792634033607451150833205281710306050584869]
55, [-19787331441001612103854548589972560860344628543488, 458747643022697229679034253322268753530821000822784, -498322305904700479352353139318239295763736705069711, 25465609788816025420512226447159951]
57, [-8229610565782363379309230843178332823243893220736032646620365455360, 190794522308574764832610713856918040956153313967530080944929405665280, -207253743440139491819646759246409433120794643349962120562288056908221, 228417583703134726518153542940959974324117920332221]
59, [31760943205719479335823285328215456670357182181431056531456, -736342739254385809841346747190949908172635592872774271500288, 799864625665824771067695390146684566835478274489741411197277, -2554702072299303745322128317254105026634077]
61, [11228750712595691857993899010018102022037871254939583265845661500073574400, -260326307205776478317777039351824473045960872891225249332788909477388615680, 282783808630967862053243614724676254536783858288976645470904467362872436199, -19478777352815577446747242887872229606424234885336671719]
63, [-4420734755976561477819352219602229305413295023063050334765056, 102489901469508664033765456187485011871537248089030410809376768, -111331370981461787011628115392902787866895877725893259127542497, 22223954766213317384532039590736747648635617]
65, [-8683784727375452650614759128937062365289209832692753414899240041785839908290560, 201324053628847266745608051881163101313191326264291637750219224169082169579274240, -218691623083598172047779067967099055178898618094875773172951473280049980872394819, 941497828061829135128872165198628592933769027396131476539459]
67, [-14974422008672519045548749269274479895709052562995131254761848832, 347165600504945911039907687335075786243749800574965856300859129856, -377114444522290949135710157102120959683567879802427217434372529149, 4704971228496213648399974101471098623989701629]
69, [-185314968362017705213459062217245680311133697058676732436818759208189611338841184010240, 4296324909014516204385185193530987448037813346593314121138374551245928228542091152588800, -4666954845738551614813359061035594749614058479585029726717416648664601047412933868051203, 1255743070115940953977738874362140705404579002293596193160347441923]
71, [61765379767975943763564298671677357518170611735643783882563340122256733896704, -1431962792630438802123000716647117309276993835203533818855166566376154911997952, 1555493552166390689651342233655072283397499596396297454512791444169473685730933, -1212919664600259084164537721476067892498197548805305939573]
73, [24938099006978845638493179864064412055475176319643422932932502097890221096955660601267322880, -578162556938456821207869671616593770935223736916678380623777686489311432178114313091557621760, 628038754952414512484866593040473963109200116123894385538012001016336947709809541332345736249, -10561695751368063026026567929159048369310331245073337783907038253468729]
75, [-390204487785772245552169373144049299002675650800864945178530088720281894912, 9046464380622959539308570407853288465993852509926942216097125384641855881216, -9826873356194504030413388069978046446128816575369031099273417647506967481289, 478915836659382129612763840358992819232085424547810249]
77, [33039031803780971646949418274018429138835113108412595794105156149033692062506396416263621777786142720, -765973825875797056544315681412317221875941036687818278628814248110123988836413433976388600828012789760, 832051889483358999838215392496244159658913985805066846948699934993472641291634017861906619893964508777, -874535890079505302722900423376731675374585281268330207791052990775510379433577]
79, [-7948573908706841534676092247545401116795900549084783397826629754111252014199698620416, 184278994713518335994789403080017025025990349547503367528394518575528037349365821472768, -200176142530932019064142197303154738396565412442419820990606363048408446558455160474447, 609728046911136983261796746886666558584964657905180689941760847]
81, [2504310589755493072033404099474008766889806532267901542178442869871759832765575608026779319275662085822873600, -58059702687678837992904172486551067085554170734728242382229349511158799691722489501521984579816048690032803840, 63068323867189824136970984828533419957022955083560281658006829607403557849050730415902260092229677419061847917, -4143034335337689171284296236191420594356501238491797089698326716873862304557383296877]
83, [-1395026488446729377451315071249881234067003497677540212504033354616854984062500885760049152, 32342163744378698719941090436550869833177196878205557958055175630121978659907869773928595456, -35132216721272157474843727267253336642684999570713820055790394625517711935315503090798526799, 6688202704341373795697153181672727152286162023307282631545349833039]
85, [121834497350173728318760510741188576953548165930928245410766168786624862797993941252308201151758207273972877230080, -2824599601259728750686995912605218413540431435651911040392709005764727911440879191392092816100292157242659171205120, 3068268595960076207324516946684987335955360566222912476729629534499825775961375644019214528685521313367393538495207, -12597391768507832798709144945515388191161848138924508570122505310281712741576788612829927]
87, [19987577845270390472333484509168881791283688521932190774797721782573154748703807062087127007232, -463390136946446960482794719798163922150023456255553446904699367701772368255311232229966511865856, 503365292636987741427461694805678831519987364417154860763793978244275335656845323444057212840399, -5989177145787396531117737032309499166977356657904126477089916446960079]
89, [-11382117140085825344388010664609015374285519312312917282482536944004293264331882522495597719244213824649373936999297187840, 263881940128775051515996279378134951586121054780315800597831250209783986449098272472927097491332545067810091221960255078400, -286646174408946702204772300780908190366644265495400093696686323143306140300716829134579716140753227988482849620673360747903, 73555208031952172090458458533889999045513567322954791616661423210932255271374010524714511293823]
91, [1522102149490661104692353150683889113184002478291883954809768441239186413856991721498790709479797161984, -35288273994933232485916603409966853301095862799681426584759584410796993537976853479539612667235576840192, 38332478293914554695301309739840272249209910983298299400567223489653685161026294986622032987889620762143, -28505640721746043227033104906188102196378318795335458064084838901694449597983]
93, [-491241796584098093801513455786168313619673947101988037736189133974977583423469763874264174601567675405782707819514255149013, 9602971205724114176441661032518594126103795203741664904588953964555593171392107534391107297007978677705326914518669298738292, 30735077653585262005264092518337151637958462294631864036348466652940038043489268227002700528578448453149334294317101287693358, 3855887201060467794813314593482421805307176414738677915854759562618599349548711551429633075059624382748270074420815607856026]
95, [-52673654030825340292175962658467915662819993918864806578106026195130784352819264976035664886452866573194493952, 1221181072752626302938026487122610613291421157043598899505831940516863985869032018194363278902262562813416308736, -1326528380814276983522378412501200333032446940726844360488172383974157790016379484551078354355314559831909837369, 61653888415385795845515847826128391067032235441708936404643745680146263872104540729]
97, [456364510414326437385692885329843311955447838091850107200649671557377948784823766367805778178813664916797029021868927303217, -10443267668689027599579320122734359006139100166561518160777468720377586339501072469847377838903615374771917413853879389405317, 8205784822378328917683958226933549714084739188875760024899097891051344144656212052476817229825990022513903291071465656504472, -11316027937409660831726488097960791692249310808175202129805052771789724886327638934726813571755642386646111938070680297622299]
99, [-8155244689141447419499275678028996134604225751254697210874866778006303671551717424891658961349146880033746321408, 189070430773945373364910234480616270780118860471770395820007862860130191605543975285496541228248415440837420253184, -205380920152228268203908785837270864134010756753913905817577091508686961213707590513865217055397647954008200549611, 596601084683444779634115575819495092544162265060180378585357904450938753103287653611]
101, [-1554274346397441479432503923086811022876576252166888256451590872704190489115389632727750879307230104170590339219560490821330, 33956109539951654920292462565614977022098845388860600985659194615550241140221285385301146702411685813055542107241178251853270, 11016717056318325203555285106030467252029474415240273554727712214806804598419485114710287964235308576291996992670136150357855, 3191095482244020780607369935557218962513586714375459427428762978640964967057601171890655510443821356053761294742847109979621]
103, [1118824616252685649934065805500145702719461484493459585739567465074894257346035969136797471482612246119200476677275648, -25938725350208855527021822720889075222130762005597312373964115654492993255833638599159511480099565076669087605948678144, 28176374582714226826889954331894482139159696215641678874815017376262537833500068916879975020899681768174515731996583601, -5115511590011241057447329371766791619756062974358379446868597834892199267027172693354161]
105, [-176710837375537193816242358548002715816616876314510188934982163046193017543813702752576463652761090265025033547820207134067, 4133587126218477834598952595118638144396998175685455833678178574144011398834588857104945924874279141552052785497156577822591, -5317916571666089386029817433599380529013528940015974648376589365873582980379497114125406016317853544053925879646619424913526, -10339103646093565843157198391472641859698418134105422347226738703340140847888203934883537370469210381757327017307923172792940]
107, [-368405925295638437222037141558011197109559938049757486928246128085803176236523403788285735920239751540790486101139504385846, 8129559352144475644522725865432736283708608674226550501981527473562701796853359787899102577642841455207707204962890485300412, 646213711272239871177306970701659509022694062986607507621169230487099514291498469133750386256848948794888449449780276427493, 5625461083064419151836355294224799739700680406988346931514477506460187975609157936200669001667740991143693310926233800278618]
109, [606171310404229167000162438730244946541623366625293588945808661752526980388523594299811481436566342489430985994364263656850, -12858490332310573654134288861658687105538374245428572068823729917640832083201340473308943359024949952246360143459037035188791, -13576238875722345225346972567315986396744655167145305276502569793239531682415248852847350572195249016500516958055098754731428, -2363524705816573982526678929534377357401062559018149066745347964351908730958169977255937468569199056208633676248235912509368]
111, [-25037993906775741191501700437801988670760810014291300916810723293008378911790800624156631494065591360832274046962089041291, 177251981931029916150008413027835267944627888235234469263149243157025712127533432100388271512267762510284150040172359304003, 9092631294070104815519814478605966967237697154270492060464464408487704207819579930970906617093659726016043152050968901593047, 5891700841428845823875377503019423421328757381398094532874182293953670149800248755289939281845990702409649382901559000828056]
113, [111504144939795443387356327963339053259143153292248584301604826478126768875784129117042862245157224302945489426632915785596, -1772224532929303793737838290017831918405872116417789435138087234787994292241698215457719455021073062032155774601221481871375, -16820265970466849927646787154365990821678176856068200353343889083362300804322539586173206371449485057351334053759627828082504, 2660284776738350850560113216379300640370658540912911685266371451683412701802465054039964814417654219626522715956377765307690]
115, [187371534382793133845834866664381719197788900204121130977239808037523734695041161427051212508421106846182924535628432774467, -4399769199695083825881170530080333750682463274124166779234658501311406825461455514400768395974915294299106842332728827257791, 6052917182861016413905068151181370823263387579215446536689507383116722647592369454623872525923166083349653959718066287642154, 6457809969052594830583792627547044454565178523821552193505214669153868614162151505421379486214802373432326270878740518075143]

Results with Pi^z, S(z,1), S(z,2), S(z,4).
The display is the integer relation that
make the vector [Pi^z, S(z,1), S(z,2), S(z,4)] equal to 0.

3, [-1, 720, -900, 180]
5, [1, -7056, 6993, 63]
7, [13, -907200, 921375, -14175]
9, [164, -112907520, 112920885, -13365]
11, [-32072, 217945728000, -218158565625, 212837625]
13, [76192, -5109989068800, 5109979245525, 9823275]
15, [-219824, 145508493312000, -145517374445625, 8881133625]
17, [-187296448, 1223606598230016000, -1223606634625249875, 36395233875]
19, [-3352363136, 216153986111078400000, -216154810673098171875, 824562019771875]
21, [-71277685379584, 45359209940997897584640000, -45359209919865550963629375, -21132346621010625]
23, [404527384477184, -2540736205097615504179200000, 2540736810856331395448765625, -605758715891269565625]
25, [-2296123608713216, 142333345151212060346941440000, -142333345152247545331370960625, 1035484984429520625]
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References
[1] Berndt, Bruce,  Ramanujan Notebooks (volumes I to V), Springer Verlag.

[2] Plouffe, Simon, Identities inspired from Ramanujan Notebooks II, 1998, unpublished notes.

[3] Finch, Steven R., Mathematical Constants, Encyclopedia of Mathematics and its Applications, 94 (2003).

[4] Flajolet, Philippe. Recent articles from INRIA.

I take the occasion to thank Bruce Berndt for the remarkable piece of work he did with these collected works of S. Ramanujan.