Atal's Integrals
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This page contains the values of the following integral as \(N\) varies:
$$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_5(t).$$
Here \(\zeta\) is the Riemann zeta function.
For \(N=1\):
$$\frac{4}{5}-\frac{\pi ^6}{1134}$$
For \(N=2\):
$$\frac{421}{320}-\frac{\pi ^6}{1134}$$
For \(N=3\):
$$\frac{273469}{233280}-\frac{\pi ^6}{1134}$$
For \(N=4\):
$$\frac{8172853}{7464960}-\frac{\pi ^6}{1134}$$
For \(N=5\):
$$\frac{9503998733}{9331200000}-\frac{\pi ^6}{1134}$$
For \(N=6\):
$$\frac{46179076609}{46656000000}-\frac{\pi ^6}{1134}$$
For \(N=7\):
$$\frac{5152765711072241}{5489031744000000}-\frac{\pi ^6}{1134}$$
For \(N=8\):
$$\frac{325136172634357799}{351298031616000000}-\frac{\pi ^6}{1134}$$
For \(N=9\):
$$\frac{226580272298654091721}{256096265048064000000}-\frac{\pi ^6}{1134}$$
For \(N=10\):
$$\frac{44966356020187087349}{51219253009612800000}-\frac{\pi ^6}{1134}$$
For \(N=11\):
$$\frac{76367947522978007398690877}{90738031080962661580800000}-\frac{\pi ^6}{1134}$$
For \(N=12\):
$$\frac{76140609289457074351450877}{90738031080962661580800000}-\frac{\pi ^6}{1134}$$
For \(N=13\):
$$\frac{352738190926129436445403153201493}{437975145063870303582159667200000}-\frac{\pi ^6}{1134}$$
For \(N=14\):
$$\frac{352266907771366247470290999601493}{437975145063870303582159667200000}-\frac{\pi ^6}{1134}$$
For \(N=15\):
$$\frac{338665388011717711152424348508501}{437975145063870303582159667200000}-\frac{\pi ^6}{1134}$$
For \(N=16\):
$$\frac{108366622462411291572994259250780349}{140152046420438497146291093504000000}-\frac{\pi ^6}{1134}$$
For \(N=17\):
$$\frac{2512419376433100610523371457312467204219081}{3382929690964537235124904363538251776000000}-\frac{\pi ^6}{1134}$$
For \(N=18\):
$$\frac{839339328530217198466106559306798098473027}{1127643230321512411708301454512750592000000}-\frac{\pi ^6}{1134}$$
For \(N=19\):
$$\frac{113705572855806958153771402596742957160904059455361}{159152907672484393983755270823401332001734656000000}-\frac{\pi ^6}{1134}$$
For \(N=20\):
$$\frac{113969449688069032913835991869203445521462507089793}{159152907672484393983755270823401332001734656000000}-\frac{\pi ^6}{1134}$$
For \(N=21\):
$$\frac{21887808276952233925337625675751569382354806396493}{31830581534496878796751054164680266400346931200000}-\frac{\pi ^6}{1134}$$
For \(N=22\):
$$\frac{21946391520390033745028260614657587072124330876493}{31830581534496878796751054164680266400346931200000}-\frac{\pi ^6}{1134}$$
For \(N=23\):
$$\frac{3114078250984007866513393094995737802911062567906429817277}{4712068434846229620402292614955595637332187868613836800000}-\frac{\pi ^6}{1134}$$
For \(N=24\):
$$\frac{3128422774349548430571117207931895385510365839138044657277}{4712068434846229620402292614955595637332187868613836800000}-\frac{\pi ^6}{1134}$$
For \(N=25\):
$$\frac{233934532637853441078653386917683366704662964717128730744853369}{368130346472361689093929110543405909166577177235456000000000000}-\frac{\pi ^6}{1134}$$
For \(N=26\):
$$\frac{234880996294128469035909130665181810758075285667673613056193401}{368130346472361689093929110543405909166577177235456000000000000}-\frac{\pi ^6}{1134}$$
For \(N=27\):
$$\frac{4202493957989099942490181847132741278552129379994131474819615111}{6881205707137222342294213374003664302113711851401216000000000000}-\frac{\pi ^6}{1134}$$
For \(N=28\):
$$\frac{12662732068011312927747457971072626658698504552510588424458845333}{20643617121411667026882640122010992906341135554203648000000000000}-\frac{\pi ^6}{1134}$$
For \(N=29\):
$$\frac{7190938811748502295105458973834962462378096532725964611069350121632410893}{12279304893610547989076528274622423999057276209262589413675008000000000000}-\frac{\pi ^6}{1134}$$
For \(N=30\):
$$\frac{7234852526385969124015304227409824559921060434093245872300040022560410893}{12279304893610547989076528274622423999057276209262589413675008000000000000}-\frac{\pi ^6}{1134}$$