Atal's Integrals
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This page contains the values of the following integral as \(N\) varies:
$$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_8(t).$$
Here \(\zeta\) is the Riemann zeta function.
For \(N=1\):
$$\frac{7}{8}-\frac{8 \zeta (9)}{9}$$
For \(N=2\):
$$\frac{705}{512}-\frac{8 \zeta (9)}{9}$$
For \(N=3\):
$$\frac{4087649}{3359232}-\frac{8 \zeta (9)}{9}$$
For \(N=4\):
$$\frac{1465799497}{1289945088}-\frac{8 \zeta (9)}{9}$$
For \(N=5\):
$$\frac{10828231325571599}{10077696000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=6\):
$$\frac{10504876353139727}{10077696000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=7\):
$$\frac{15149726775272481831307}{15061903105536000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=8\):
$$\frac{7625433237710398375363559}{7711694390034432000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=9\):
$$\frac{3942239366840473534515874221019}{4098310578334288576512000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=10\):
$$\frac{779902568227135736771655418463}{819662115666857715302400000000}-\frac{8 \zeta (9)}{9}$$
For \(N=11\):
$$\frac{816362321370024636803940155994574722811}{878509269562200943192195203072000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=12\):
$$\frac{8913260724598949013740378844399321950921}{9663601965184210375114147233792000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=13\):
$$\frac{92512018561800471220256719122626080587458881272533}{102477660980717526752398069144176948412416000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=14\):
$$\frac{92009768586357839472480074823143834646722881272533}{102477660980717526752398069144176948412416000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=15\):
$$\frac{90173649224899175386261269553645321200336562109141}{102477660980717526752398069144176948412416000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=16\):
$$\frac{45979958680173259581173213293106037285796494516932959}{52468562422127373697227811401818597587156992000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=17\):
$$\frac{5347614404435053693250486391415802110121281074394672507179358373}{6222135400490376172009817969792472291863515560646017024000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=18\):
$$\frac{5331033098614724688008985777266149984158683819835829627179358373}{6222135400490376172009817969792472291863515560646017024000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=19\):
$$\frac{3290160822852893567792071081100521407822681394507158962803739152806544159}{3913852919402447630782703820774262476152213625633929215283822592000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=20\):
$$\frac{62358261962515386756111139057423189249606668704265805771624411557258378061}{74363205468646504984871372594710987046892058887044655090392629248000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=21\):
$$\frac{489617688177547872955535810707291334792825648895596477669743799706382097}{594905643749172039878970980757687896375136471096357240723141033984000000}-\frac{8 \zeta (9)}{9}$$
For \(N=22\):
$$\frac{44422576167670446662398930783001223781410109635961652954542934704943827}{54082331249924730898088270977971626943194224645123385520285548544000000}-\frac{8 \zeta (9)}{9}$$
For \(N=23\):
$$\frac{863859485686372414977011751582803310270667482136039644459789303323408657886557027911}{1071515883558180549230673354397452914854378805175041617652716440093387330158592000000}-\frac{8 \zeta (9)}{9}$$
For \(N=24\):
$$\frac{862542043467868299079789229054386757280177925010419653016910985077556291694853027911}{1071515883558180549230673354397452914854378805175041617652716440093387330158592000000}-\frac{8 \zeta (9)}{9}$$
For \(N=25\):
$$\frac{206688956329256029447135003421211047682379267038565915521865007997569519220685961756651844727}{261600557509321423152019861913440653040619825482187894934745224632174641152000000000000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=26\):
$$\frac{206425372715619880570925507303369484968314830851946574737696395762401495208073454306464134263}{261600557509321423152019861913440653040619825482187894934745224632174641152000000000000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=27\):
$$\frac{107675279993547241796372206664184322171561782846875352009801781003128452979428630630081605976862983}{139025261883311286441332587435140814092560040674079417072015936923747523468460032000000000000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=28\):
$$\frac{107564132447864993099970727159942363187764331737562079134611080032635531665023000630081605976862983}{139025261883311286441332587435140814092560040674079417072015936923747523468460032000000000000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=29\):
$$\frac{1531333691932717197832356109840719758930308897748473810823160884818746460240245307909992488173581935156837357227}{2016859768474613201346047627363556653202243038957242177552417297914779616188424590571262967808000000000000000000}-\frac{8 \zeta (9)}{9}$$
For \(N=30\):
$$\frac{1530163966550149844148551750291933924292822475742732950203619152563468823960919552831549730218357758260837357227}{2016859768474613201346047627363556653202243038957242177552417297914779616188424590571262967808000000000000000000}-\frac{8 \zeta (9)}{9}$$