Atal's Integrals
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This page contains the values of the following integral as \(N\) varies:
$$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_{11}(t).$$
Here \(\zeta\) is the Riemann zeta function.
For \(N=1\):
$$\frac{10}{11}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=2\):
$$\frac{63499}{45056}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=3\):
$$\frac{29801386843}{23944605696}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=4\):
$$\frac{57030789837367}{49038552465408}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=5\):
$$\frac{26497507740710325709259}{23944605696000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=6\):
$$\frac{25718070928036821727691}{23944605696000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=7\):
$$\frac{345755427555203598098958145582891}{331424164353036496896000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=8\):
$$\frac{1392504827471666976983875022559474661}{1357513377190037491286016000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=9\):
$$\frac{724774317266078784003733856096093738160251}{721438266687250714406531629056000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=10\):
$$\frac{716928822291339669800902771287040464048251}{721438266687250714406531629056000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=11\):
$$\frac{201159469039196445600693388522662891078157393153956577}{205834757111243649981930784496178708873216000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=12\):
$$\frac{199634859884300677635769315734863807812252393153956577}{205834757111243649981930784496178708873216000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=13\):
$$\frac{4584768790740196301653151760367159252871156488889262233144800507537}{4795555692342955898731505374745518150394905032860368896000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=14\):
$$\frac{4559037035933602761869951778440440981307966953166114969144800507537}{4795555692342955898731505374745518150394905032860368896000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=15\):
$$\frac{4500487244076160283291148789612564726242056264665202062308135597713}{4795555692342955898731505374745518150394905032860368896000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=16\):
$$\frac{18354268410557642383527782016817092790717662076050222219166885015611973}{19642596115836747361204246014957642344017531014596070998016000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=17\):
$$\frac{10566083893685624202813441775974856385598035853641713040849709181782698106912303215953}{11444213294009419400559431737590605811360100349544654479719512350367154176000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=18\):
$$\frac{3509938231675954263076443339788309338085025730802649888757850165667772456970767738651}{3814737764669806466853143912530201937120033449848218159906504116789051392000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=19\):
$$\frac{23042364172144496443538465987050898873258625934691297417349135092611697831814994436596646595877666433}{25329648020606341881531184407931793519722141856069694215889603505505100121926593031438336000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=20\):
$$\frac{22977916224618070120994796142367539567414721240698343435838808936570381482453436623298796201081306753}{25329648020606341881531184407931793519722141856069694215889603505505100121926593031438336000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=21\):
$$\frac{22728176864194903986769812317083975378198233681247640426819995440116630320570421987850667050612293249}{25329648020606341881531184407931793519722141856069694215889603505505100121926593031438336000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=22\):
$$\frac{22675413878954171856390255222625079776945614814704948343297498329210109993010115288524203050612293249}{25329648020606341881531184407931793519722141856069694215889603505505100121926593031438336000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=23\):
$$\frac{491670715635276708059701905407694889535155471805975485865944972409817162945843959217756814503105455827571551979112929}{555089723366854903028574884869334132055235705746509022002960579852696006265414063696857172504017010425856000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=24\):
$$\frac{490710218722338759160201051727274048061533390597050627569547485482720136470032977124456159679262505331290551979112929}{555089723366854903028574884869334132055235705746509022002960579852696006265414063696857172504017010425856000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=25\):
$$\frac{118561518987750056507529969349887571054309017011990870616323778606554155310928950319408613868718441488441209311743423359367729}{135519951993861060309710665251302278333797779723268804199941547815599610904642105394740520630863527936000000000000000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=26\):
$$\frac{118363183229312860304578234168438397831558868760687249936040626446398856110495070638236748012686899306019496149479292135946801}{135519951993861060309710665251302278333797779723268804199941547815599610904642105394740520630863527936000000000000000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=27\):
$$\frac{20753737375095591032637832657350915202201859696293119708638267417856858133960071131688436079109792560279429684391808164006567956747}{24006952935856505250684315217272444699997275284637898857607045370890024272924635044362099008195581383278592000000000000000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=28\):
$$\frac{62171181322445654104957883914854427784281081281544970902414419852963739489593080543435377920654063669219603053175424492019703870241}{72020858807569515752052945651817334099991825853913696572821136112670072818773905133086297024586744149835776000000000000000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=29\):
$$\frac{21774786250666937104720747202568735188573524103511228515164929665894996729470497778484447485811826915376209278275785810023063837578922232025406708881}{25482044545271903764415406712717256144531135710978182819683207303100370866569475382199155385547923365933656312855725202210816000000000000000000000000}-\frac{691 \pi ^{12}}{696559500}$$
For \(N=30\):
$$\frac{21747399155813255693167590925644611056429102567603986933864081323826733057036080624580056312429448366948473118782431211504792619108243704025406708881}{25482044545271903764415406712717256144531135710978182819683207303100370866569475382199155385547923365933656312855725202210816000000000000000000000000}-\frac{691 \pi ^{12}}{696559500}$$