Atal's Integrals
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This page contains the values of the following integral as \(N\) varies:
$$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_3(t).$$
Here \(\zeta\) is the Riemann zeta function.
For \(N=1\):
$$\frac{2}{3}-\frac{\pi ^4}{120}$$
For \(N=2\):
$$\frac{1}{240} \left(295-2 \pi ^4\right)$$
For \(N=3\):
$$\frac{7285-54 \pi ^4}{6480}$$
For \(N=4\):
$$\frac{55045-432 \pi ^4}{51840}$$
For \(N=5\):
$$\frac{12480829}{12960000}-\frac{\pi ^4}{120}$$
For \(N=6\):
$$\frac{12298861}{12960000}-\frac{\pi ^4}{120}$$
For \(N=7\):
$$\frac{27178152761}{31116960000}-\frac{\pi ^4}{120}$$
For \(N=8\):
$$\frac{435919593551}{497871360000}-\frac{\pi ^4}{120}$$
For \(N=9\):
$$\frac{32822486258881}{40327580160000}-\frac{\pi ^4}{120}$$
For \(N=10\):
$$\frac{33037492927681}{40327580160000}-\frac{\pi ^4}{120}$$
For \(N=11\):
$$\frac{449605470380348977}{590436101122560000}-\frac{\pi ^4}{120}$$
For \(N=12\):
$$\frac{458246492642738977}{590436101122560000}-\frac{\pi ^4}{120}$$
For \(N=13\):
$$\frac{12123635045343990942097}{16863445484161436160000}-\frac{\pi ^4}{120}$$
For \(N=14\):
$$\frac{12337789390241074302097}{16863445484161436160000}-\frac{\pi ^4}{120}$$
For \(N=15\):
$$\frac{11536979756181145990993}{16863445484161436160000}-\frac{\pi ^4}{120}$$
For \(N=16\):
$$\frac{187556263065081551434093}{269815127746582978560000}-\frac{\pi ^4}{120}$$
For \(N=17\):
$$\frac{14530147643148852388843853953}{22535229284522356952309760000}-\frac{\pi ^4}{120}$$
For \(N=18\):
$$\frac{4977645748996631704143044651}{7511743094840785650769920000}-\frac{\pi ^4}{120}$$
For \(N=19\):
$$\frac{1796755468556268036694547693608913}{2936813615588238080381960232960000}-\frac{\pi ^4}{120}$$
For \(N=20\):
$$\frac{1840256417469776798115090474592913}{2936813615588238080381960232960000}-\frac{\pi ^4}{120}$$
For \(N=21\):
$$\frac{1709372097447491317129365026420369}{2936813615588238080381960232960000}-\frac{\pi ^4}{120}$$
For \(N=22\):
$$\frac{249419318347486103743442819505767}{419544802226891154340280033280000}-\frac{\pi ^4}{120}$$
For \(N=23\):
$$\frac{64286130836661746214637545321323583047}{117405836999975447521738304793108480000}-\frac{\pi ^4}{120}$$
For \(N=24\):
$$\frac{66530313772092388211432330819249793047}{117405836999975447521738304793108480000}-\frac{\pi ^4}{120}$$
For \(N=25\):
$$\frac{38006827534795298420252213008762230297287}{73378648124984654701086440495692800000000}-\frac{\pi ^4}{120}$$
For \(N=26\):
$$\frac{39045463658374546070236240019782452127943}{73378648124984654701086440495692800000000}-\frac{\pi ^4}{120}$$
For \(N=27\):
$$\frac{971176876585950022576920142667184007454461}{1981223499374585676929333893383705600000000}-\frac{\pi ^4}{120}$$
For \(N=28\):
$$\frac{20968922107876207087477945499550279556543681}{41605693486866299215516011761057817600000000}-\frac{\pi ^4}{120}$$
For \(N=29\):
$$\frac{13465342652187303547173131772682540741205894121361}{29426916495084282975449380314372734289945600000000}-\frac{\pi ^4}{120}$$
For \(N=30\):
$$\frac{14015684254600423896008500471333812044599899081361}{29426916495084282975449380314372734289945600000000}-\frac{\pi ^4}{120}$$