Atal's Integrals
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This page contains the values of the following integral as \(N\) varies:
$$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_6(t).$$
Here \(\zeta\) is the Riemann zeta function.
For \(N=1\):
$$\frac{5}{6}-\frac{6 \zeta (7)}{7}$$
For \(N=2\):
$$\frac{515}{384}-\frac{6 \zeta (7)}{7}$$
For \(N=3\):
$$\frac{111001}{93312}-\frac{6 \zeta (7)}{7}$$
For \(N=4\):
$$\frac{9946927}{8957952}-\frac{6 \zeta (7)}{7}$$
For \(N=5\):
$$\frac{20392187373833}{19595520000000}-\frac{6 \zeta (7)}{7}$$
For \(N=6\):
$$\frac{19790336838281}{19595520000000}-\frac{6 \zeta (7)}{7}$$
For \(N=7\):
$$\frac{105988340950329589}{109780634880000000}-\frac{6 \zeta (7)}{7}$$
For \(N=8\):
$$\frac{93446693282509296119}{98363448852480000000}-\frac{6 \zeta (7)}{7}$$
For \(N=9\):
$$\frac{590754900021048935274259}{645362587921121280000000}-\frac{6 \zeta (7)}{7}$$
For \(N=10\):
$$\frac{585022607406258694149907}{645362587921121280000000}-\frac{6 \zeta (7)}{7}$$
For \(N=11\):
$$\frac{11022655186805796780358306541177}{12576291107821424895098880000000}-\frac{6 \zeta (7)}{7}$$
For \(N=12\):
$$\frac{10958820758616649583991856541177}{12576291107821424895098880000000}-\frac{6 \zeta (7)}{7}$$
For \(N=13\):
$$\frac{666017446412833044790718282452046184509}{789143616376081512994335288360960000000}-\frac{6 \zeta (7)}{7}$$
For \(N=14\):
$$\frac{663431291558633049049335040802126184509}{789143616376081512994335288360960000000}-\frac{6 \zeta (7)}{7}$$
For \(N=15\):
$$\frac{128679768065578449985098917956064333017}{157828723275216302598867057672192000000}-\frac{6 \zeta (7)}{7}$$
For \(N=16\):
$$\frac{82147589280089861881402269893896991847799}{101010382896138433663274916910202880000000}-\frac{6 \zeta (7)}{7}$$
For \(N=17\):
$$\frac{32691627658306362414469237555717161074254604474477}{41448466476823341693686758239117909939978240000000}-\frac{6 \zeta (7)}{7}$$
For \(N=18\):
$$\frac{32656243664077101595495596059106141654770444474477}{41448466476823341693686758239117909939978240000000}-\frac{6 \zeta (7)}{7}$$
For \(N=19\):
$$\frac{496637798737625977350440269319809630523945989720637493079}{649993207167040063781175226464436912001644473876480000000}-\frac{6 \zeta (7)}{7}$$
For \(N=20\):
$$\frac{377205585777920762107480075893196160614641623326921832893}{493994837446950448473693172112972053121249800146124800000}-\frac{6 \zeta (7)}{7}$$
For \(N=21\):
$$\frac{1829413930349675921715284868235744859525530652978792505521}{2469974187234752242368465860564860265606249000730624000000}-\frac{6 \zeta (7)}{7}$$
For \(N=22\):
$$\frac{1829136666985157552349189147740228456133508795507304505521}{2469974187234752242368465860564860265606249000730624000000}-\frac{6 \zeta (7)}{7}$$
For \(N=23\):
$$\frac{6036606712521342055982973841952050538647762506763560074617700792887}{8409830966130026997556464112401990026335335479905950187388928000000}-\frac{6 \zeta (7)}{7}$$
For \(N=24\):
$$\frac{6041344272595782457759980431237766283374955522082918281970270792887}{8409830966130026997556464112401990026335335479905950187388928000000}-\frac{6 \zeta (7)}{7}$$
For \(N=25\):
$$\frac{91442123062836070109189319736017611481603507590660576676046291570474351}{131403608845781671836819751756281094161489616873530471677952000000000000}-\frac{6 \zeta (7)}{7}$$
For \(N=26\):
$$\frac{2287827954688105105837738046334410480109481977576010790068542840464375767}{3285090221144541795920493793907027354037240421838261791948800000000000000}-\frac{6 \zeta (7)}{7}$$
For \(N=27\):
$$\frac{14538557849097091710190381548977584645768734064394623381854031376286769407287}{21553476940929338723034359781824006469838334407680835616976076800000000000000}-\frac{6 \zeta (7)}{7}$$
For \(N=28\):
$$\frac{14553916420465646549077332850514644859844169459413922446644565176286769407287}{21553476940929338723034359781824006469838334407680835616976076800000000000000}-\frac{6 \zeta (7)}{7}$$
For \(N=29\):
$$\frac{242888394719495991858459642920608441720886131240129107109168583191807620773800713263683}{371794811259914792481506715393468338011527007759073193942599025412084531200000000000000}-\frac{6 \zeta (7)}{7}$$
For \(N=30\):
$$\frac{243325111189662262056861393037055165840813441932594519326993519942059280722843913263683}{371794811259914792481506715393468338011527007759073193942599025412084531200000000000000}-\frac{6 \zeta (7)}{7}$$