Atal's Integrals
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This page contains the values of the following integral as \(N\) varies:
$$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_{10}(t).$$
Here \(\zeta\) is the Riemann zeta function.
For \(N=1\):
$$\frac{9}{10}-\frac{10 \zeta (11)}{11}$$
For \(N=2\):
$$\frac{14341}{10240}-\frac{10 \zeta (11)}{11}$$
For \(N=3\):
$$\frac{83088421}{67184640}-\frac{10 \zeta (11)}{11}$$
For \(N=4\):
$$\frac{1073118722159}{928760463360}-\frac{10 \zeta (11)}{11}$$
For \(N=5\):
$$\frac{438082368615857773189}{399076761600000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=6\):
$$\frac{425128573540406530693}{399076761600000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=7\):
$$\frac{271724453576313718123706407633}{263035051073838489600000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=8\):
$$\frac{547120008047922806155634365410509}{538695784599221226700800000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=9\):
$$\frac{284295962532777415009517407910352875969}{286285026463194727939099852800000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=10\):
$$\frac{281201343484342836303512497626077688257}{286285026463194727939099852800000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=11\):
$$\frac{7160698065855959207271024058197562117107544300697}{7425496288284402957501110551810198732800000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=12\):
$$\frac{7106218831615443167765359715478651793407544300697}{7425496288284402957501110551810198732800000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=13\):
$$\frac{964017033616177970577673897698578766366858137559703779518753}{1023667719533235119310498342415174174200778928947200000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=14\):
$$\frac{12461692954902609414000594823655648962860376178368949133743789}{13307680353932056551036478451397264264610126076313600000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=15\):
$$\frac{2456026034208111717212338451884775922014848339589377732478089}{2661536070786411310207295690279452852922025215262720000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=16\):
$$\frac{25040869719202221459350379174886180641084988742917737945365347399}{27254129364852851816522707868461597213921538204290252800000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=17\):
$$\frac{846447001627497101553103219348891538981904705689822892648761966164661947740317}{934050695547052571657054688578627729407855245207786860175732139622400000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=18\):
$$\frac{843552604635849568342367106732535111456131541842534094314926869281461947740317}{934050695547052571657054688578627729407855245207786860175732139622400000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=19\):
$$\frac{32325873457698015570909778067137505710702454660189161260350178587402083500259030389715265141}{36269269116112562305157267351287334268360555073967561815391356014563781766230835200000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=20\):
$$\frac{257887835240736457411202401164055485782347114826512216171111816641526326988216158725019481}{290154152928900498441258138810298674146884440591740494523130848116510254129846681600000000}-\frac{10 \zeta (11)}{11}$$
For \(N=21\):
$$\frac{6365289306214301362518921995621502467317389033838027488647702178530668109093258114386184113}{7253853823222512461031453470257466853672111014793512363078271202912756353246167040000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=22\):
$$\frac{6350651828624985020476953177511054674979222037721479194290737985737985609575318402386184113}{7253853823222512461031453470257466853672111014793512363078271202912756353246167040000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=23\):
$$\frac{5975639507981521982089741026835568229086171783392011858963166767773217805744940714476975060913492628841751}{6911542705247655918021132296325916895994874562858421323258293730278436238801133254217018984366080000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=24\):
$$\frac{5964164648754895123506699710326253227059978265865856443621567869101996358786332068007688481299736628841751}{6911542705247655918021132296325916895994874562858421323258293730278436238801133254217018984366080000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=25\):
$$\frac{287644596056701004890970582158144998868990771718859772641839778536037892079757912552757192225975848948695912730331}{337477671154670699122125600406538910937249734514571353674721373548751769472711584678565380096000000000000000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=26\):
$$\frac{1435863509164928532476840983585330773694771167017190738565837246525851034387558706296397875552738744723562114320967}{1687388355773353495610628002032694554686248672572856768373606867743758847363557923392826900480000000000000000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=27\):
$$\frac{753802515274255198384672829172587235182092670096578508518313581894580325788325077636849714854636395234634573596849023447}{896747355180546755060807756028257226837014680800791573841238007400610945601736586365807320817991680000000000000000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=28\):
$$\frac{752739579249331559091362467845225999432555131371277706037776634964942227753657553394286321742756175234634573596849023447}{896747355180546755060807756028257226837014680800791573841238007400610945601736586365807320817991680000000000000000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=29\):
$$\frac{9072668378625064331099571972685280512045369539282213240145701419087018228702059246108273846140800444963417679646406994183428609325572563}{10940774864251134312118178331788304789586486446719558851514684784753083290305436846036926212766386694974397449502720000000000000000000000}-\frac{10 \zeta (11)}{11}$$
For \(N=30\):
$$\frac{9061669846945998879553003702038641365937032327204434250680058439819496737365173941589816853686522455932365525494320322188139009325572563}{10940774864251134312118178331788304789586486446719558851514684784753083290305436846036926212766386694974397449502720000000000000000000000}-\frac{10 \zeta (11)}{11}$$