Atal's Integrals
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This page contains the values of the following integral as \(N\) varies:
$$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_7(t).$$
Here \(\zeta\) is the Riemann zeta function.
For \(N=1\):
$$\frac{6}{7}-\frac{\pi ^8}{10800}$$
For \(N=2\):
$$\frac{2439}{1792}-\frac{\pi ^8}{10800}$$
For \(N=3\):
$$\frac{14159207}{11757312}-\frac{\pi ^8}{10800}$$
For \(N=4\):
$$\frac{1691989787}{1504935936}-\frac{\pi ^8}{10800}$$
For \(N=5\):
$$\frac{1245279602783143}{1175731200000000}-\frac{\pi ^8}{10800}$$
For \(N=6\):
$$\frac{1208084562606247}{1175731200000000}-\frac{\pi ^8}{10800}$$
For \(N=7\):
$$\frac{956291525364723973121}{968265199641600000000}-\frac{\pi ^8}{10800}$$
For \(N=8\):
$$\frac{240719830873399836728351}{247875891108249600000000}-\frac{\pi ^8}{10800}$$
For \(N=9\):
$$\frac{1530290062971280806237992161}{1626313721561225625600000000}-\frac{\pi ^8}{10800}$$
For \(N=10\):
$$\frac{1514170259297417513562305761}{1626313721561225625600000000}-\frac{\pi ^8}{10800}$$
For \(N=11\):
$$\frac{315673648372511486975976925040141377}{348614789508809898092140953600000000}-\frac{\pi ^8}{10800}$$
For \(N=12\):
$$\frac{313474352654056875367439323340141377}{348614789508809898092140953600000000}-\frac{\pi ^8}{10800}$$
For \(N=13\):
$$\frac{249198551728125895527106854596797793102142817}{284375793597284734022638664513755545600000000}-\frac{\pi ^8}{10800}$$
For \(N=14\):
$$\frac{247957678067750365386435497784543866702142817}{284375793597284734022638664513755545600000000}-\frac{\pi ^8}{10800}$$
For \(N=15\):
$$\frac{241951616496555893922955945441308192080461153}{284375793597284734022638664513755545600000000}-\frac{\pi ^8}{10800}$$
For \(N=16\):
$$\frac{61715290204010754267052209261952670927350141333}{72800203160904891909795498115521419673600000000}-\frac{\pi ^8}{10800}$$
For \(N=17\):
$$\frac{420284695119478480025925762583897690522085405168306096353}{507836558905994020033056646767650020882998991257600000000}-\frac{\pi ^8}{10800}$$
For \(N=18\):
$$\frac{46581441732468407329679589151091792506524654851945121817}{56426284322888224448117405196405557875888776806400000000}-\frac{\pi ^8}{10800}$$
For \(N=19\):
$$\frac{6952769595148617667431396515623689447508778865854758733222995689473}{8624874212704459432000434464302453009091379853162857470361600000000}-\frac{\pi ^8}{10800}$$
For \(N=20\):
$$\frac{6939398846168343756529193515909490602563085555452968444353681580033}{8624874212704459432000434464302453009091379853162857470361600000000}-\frac{\pi ^8}{10800}$$
For \(N=21\):
$$\frac{6777947867503232228135846947775111960632314417112220450266769809409}{8624874212704459432000434464302453009091379853162857470361600000000}-\frac{\pi ^8}{10800}$$
For \(N=22\):
$$\frac{138128730101298742734073328929239875553060999958677053187746322641}{176017841075601212897968050291886796103905711289037907558400000000}-\frac{\pi ^8}{10800}$$
For \(N=23\):
$$\frac{459269401992083318451934448498610275335795367523294131210138132100749045527}{599310024420208860499066232226791941017878621885551439802473958604800000000}-\frac{\pi ^8}{10800}$$
For \(N=24\):
$$\frac{10554358854480920408141363431528195904262085550447885137552085598834728047121}{13784130561664803791478523341216214643411208303367683115456901047910400000000}-\frac{\pi ^8}{10800}$$
For \(N=25\):
$$\frac{4025254523100942091165580511745345659561547662434957081789705543754717392801693681}{5384426000650313981046298180162583845082503243503001216975351971840000000000000000}-\frac{\pi ^8}{10800}$$
For \(N=26\):
$$\frac{4022621554692247129493096601876630530846817145046561960425129671973900935089291249}{5384426000650313981046298180162583845082503243503001216975351971840000000000000000}-\frac{\pi ^8}{10800}$$
For \(N=27\):
$$\frac{2862863081853531200235447833271499033746096085833954063774499511916313781680093320521}{3925246554474078892182751373338523623065144864513687887175031587471360000000000000000}-\frac{\pi ^8}{10800}$$
For \(N=28\):
$$\frac{1262019526701778479567201184023984016744822839178382011457610878010258717720921154349761}{1731033730523068791452593355642288917771728885250536358244188930074869760000000000000000}-\frac{\pi ^8}{10800}$$
For \(N=29\):
$$\frac{616035891785107804852342202264578127308881012440409267637019661744047776320583220817772476237652321}{865943414408663461186005202841037201354910059861971993012967572392738050609250959360000000000000000}-\frac{\pi ^8}{10800}$$
For \(N=30\):
$$\frac{616034284502275874569531285991689547258885640157502941235010737347947760752187978516821833037652321}{865943414408663461186005202841037201354910059861971993012967572392738050609250959360000000000000000}-\frac{\pi ^8}{10800}$$