Atal's Integrals
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This page contains the values of the following integral as \(N\) varies:
$$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_9(t).$$
Here \(\zeta\) is the Riemann zeta function.
For \(N=1\):
$$\frac{8}{9}-\frac{\pi ^{10}}{103950}$$
For \(N=2\):
$$\frac{12809}{9216}-\frac{\pi ^{10}}{103950}$$
For \(N=3\):
$$\frac{74225321}{60466176}-\frac{\pi ^{10}}{103950}$$
For \(N=4\):
$$\frac{35498112761}{30958682112}-\frac{\pi ^{10}}{103950}$$
For \(N=5\):
$$\frac{131473758489625517}{120932352000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=6\):
$$\frac{637824915567754849}{604661760000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=7\):
$$\frac{174345623450700740342232401}{170801981216778240000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=8\):
$$\frac{175509540191912364631217462999}{174901228765980917760000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=9\):
$$\frac{10111190087363515200519406351534201}{10327742657402407212810240000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=10\):
$$\frac{2000175175343417508747439962846437}{2065548531480481442562048000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=11\):
$$\frac{10164028674105860866188136353627828129694009}{10715001858996252463926566452828569600000000}-\frac{\pi ^{10}}{103950}$$
For \(N=12\):
$$\frac{10086936752916012244343778828119360529694009}{10715001858996252463926566452828569600000000}-\frac{\pi ^{10}}{103950}$$
For \(N=13\):
$$\frac{1365250050141503482246912180507085409477028929870632641}{1477153996440454717619766727871824205195929190400000000}-\frac{\pi ^{10}}{103950}$$
For \(N=14\):
$$\frac{1357616828228795157319347603173304832100299022670632641}{1477153996440454717619766727871824205195929190400000000}-\frac{\pi ^{10}}{103950}$$
For \(N=15\):
$$\frac{6673639722995390482207465653722757059604572919011373509}{7385769982202273588098833639359121025979645952000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=16\):
$$\frac{34022973904340272243843767460746192337850542474312226972749}{37815142308875640771066028233518699653015787274240000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=17\):
$$\frac{67489192017488591575162568762392708629858946401633552624087260696551801}{76235096239304206549745572122210120713261839892769555622133760000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=18\):
$$\frac{22421285357002220996635993034882980249169657768662442770288713565517267}{25411698746434735516581857374070040237753946630923185207377920000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=19\):
$$\frac{405996865831395150144984343024744630901766115700307702655903784818669913037616849601}{467402426213009930022986138619988554638866745663618654505857352329865461760000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=20\):
$$\frac{404900121812144784177067789129516509788768484957250502519372604731874881503360852673}{467402426213009930022986138619988554638866745663618654505857352329865461760000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=21\):
$$\frac{15950386587331229039999801657277268265298497803326749497653424892158230111907212521}{18696097048520397200919445544799542185554669826544746180234294093194618470400000000}-\frac{\pi ^{10}}{103950}$$
For \(N=22\):
$$\frac{2273584269447462330351433133497766113049404358701586251128262423237643473723887503}{2670871006931485314417063649257077455079238546649249454319184870456374067200000000}-\frac{\pi ^{10}}{103950}$$
For \(N=23\):
$$\frac{92775543795116808047733541281199757599976987152985233308863151258018805446979573142130234128447}{110644867718857172383941851793539692620817422724613381914163206212834083703109402283212800000000}-\frac{\pi ^{10}}{103950}$$
For \(N=24\):
$$\frac{92607565470007855619625002043919531088194641586403526233882880156194197091623846501075914128447}{110644867718857172383941851793539692620817422724613381914163206212834083703109402283212800000000}-\frac{\pi ^{10}}{103950}$$
For \(N=25\):
$$\frac{22273038751242992666480259620278512961331820897281555331371724619057863367852349507810834183028929764847}{27012907157924114351548303660532151518754253594876313943887501516805196216579443916800000000000000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=26\):
$$\frac{22238788869726079205445658165478074485653978375642448318833431739011524022873046838255352745335225884143}{27012907157924114351548303660532151518754253594876313943887501516805196216579443916800000000000000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=27\):
$$\frac{431247729737778686161983540803320425652116911846758848560415836781316490343257582088167588086433251077586669}{531695051589420342781525260950254338343639973507950487357537692355276677130933194614374400000000000000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=28\):
$$\frac{9044408821988767135320036910117854110904225834517932821474228884244899714594973379435011549815098272629320049}{11165596083377827198412030479955341105216439443666960234508291539460810219749597086901862400000000000000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=29\):
$$\frac{3748628873267707890873361220358424110272262618596292155497008556847769075440620942030880145491575611892923467467744925029849}{4697447036385446084022174439924163482059843459584266551209279578483491407926649632074391867935765194342400000000000000000000}-\frac{\pi ^{10}}{103950}$$
For \(N=30\):
$$\frac{3744566817981883835569720417914086147114427228072631227119432916714188760022032878845864939034998968356549962275808925029849}{4697447036385446084022174439924163482059843459584266551209279578483491407926649632074391867935765194342400000000000000000000}-\frac{\pi ^{10}}{103950}$$