Atal's Integrals
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This page contains the values of the following integral as \(N\) varies:
$$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_2(t).$$
Here \(\zeta\) is the Riemann zeta function.
For \(N=1\):
$$\frac{1}{6} (3-4 \zeta (3))$$
For \(N=2\):
$$\frac{9}{8}-\frac{2 \zeta (3)}{3}$$
For \(N=3\):
$$\frac{77}{72}-\frac{2 \zeta (3)}{3}$$
For \(N=4\):
$$\frac{28}{27}-\frac{2 \zeta (3)}{3}$$
For \(N=5\):
$$\frac{199439}{216000}-\frac{2 \zeta (3)}{3}$$
For \(N=6\):
$$\frac{202103}{216000}-\frac{2 \zeta (3)}{3}$$
For \(N=7\):
$$\frac{2296827}{2744000}-\frac{2 \zeta (3)}{3}$$
For \(N=8\):
$$\frac{18975991}{21952000}-\frac{2 \zeta (3)}{3}$$
For \(N=9\):
$$\frac{12570742939}{16003008000}-\frac{2 \zeta (3)}{3}$$
For \(N=10\):
$$\frac{12922940827}{16003008000}-\frac{2 \zeta (3)}{3}$$
For \(N=11\):
$$\frac{15584558167457}{21300003648000}-\frac{2 \zeta (3)}{3}$$
For \(N=12\):
$$\frac{16463541286457}{21300003648000}-\frac{2 \zeta (3)}{3}$$
For \(N=13\):
$$\frac{32352440293194029}{46796108014656000}-\frac{2 \zeta (3)}{3}$$
For \(N=14\):
$$\frac{33889834640562029}{46796108014656000}-\frac{2 \zeta (3)}{3}$$
For \(N=15\):
$$\frac{1250096433750581}{1871844320586240}-\frac{2 \zeta (3)}{3}$$
For \(N=16\):
$$\frac{258916423867483699}{374368864117248000}-\frac{2 \zeta (3)}{3}$$
For \(N=17\):
$$\frac{1149601675070355078437}{1839274229408039424000}-\frac{2 \zeta (3)}{3}$$
For \(N=18\):
$$\frac{1226776352826146694437}{1839274229408039424000}-\frac{2 \zeta (3)}{3}$$
For \(N=19\):
$$\frac{836854812660441677371487}{1401731326612193601024000}-\frac{2 \zeta (3)}{3}$$
For \(N=20\):
$$\frac{35357579154747943123319}{56069253064487744040960}-\frac{2 \zeta (3)}{3}$$
For \(N=21\):
$$\frac{161439876170208995123923}{280346265322438720204800}-\frac{2 \zeta (3)}{3}$$
For \(N=22\):
$$\frac{168337755361694864723923}{280346265322438720204800}-\frac{2 \zeta (3)}{3}$$
For \(N=23\):
$$\frac{1844711837971040913017923141}{3410973010178111908731801600}-\frac{2 \zeta (3)}{3}$$
For \(N=24\):
$$\frac{1991475656346842467289867741}{3410973010178111908731801600}-\frac{2 \zeta (3)}{3}$$
For \(N=25\):
$$\frac{220260978336163650001419371273}{426371626272263988591475200000}-\frac{2 \zeta (3)}{3}$$
For \(N=26\):
$$\frac{1161948746305731358080821110189}{2131858131361319942957376000000}-\frac{2 \zeta (3)}{3}$$
For \(N=27\):
$$\frac{257563710828188503283510563775927}{518041525920800746138642368000000}-\frac{2 \zeta (3)}{3}$$
For \(N=28\):
$$\frac{271968823950979082417424961775927}{518041525920800746138642368000000}-\frac{2 \zeta (3)}{3}$$
For \(N=29\):
$$\frac{5859949777408931537957488666128739603}{12634514775682409397575348713152000000}-\frac{2 \zeta (3)}{3}$$
For \(N=30\):
$$\frac{6369680355134916885120879008528755603}{12634514775682409397575348713152000000}-\frac{2 \zeta (3)}{3}$$