Atal's Integrals
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This page contains the values of the following integral as \(N\) varies:
$$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_4(t).$$
Here \(\zeta\) is the Riemann zeta function.
For \(N=1\):
$$\frac{3}{4}-\frac{4 \zeta (5)}{5}$$
For \(N=2\):
$$\frac{41}{32}-\frac{4 \zeta (5)}{5}$$
For \(N=3\):
$$\frac{995}{864}-\frac{4 \zeta (5)}{5}$$
For \(N=4\):
$$\frac{67081}{62208}-\frac{4 \zeta (5)}{5}$$
For \(N=5\):
$$\frac{771851039}{777600000}-\frac{4 \zeta (5)}{5}$$
For \(N=6\):
$$\frac{752918207}{777600000}-\frac{4 \zeta (5)}{5}$$
For \(N=7\):
$$\frac{3953614491683}{4356374400000}-\frac{4 \zeta (5)}{5}$$
For \(N=8\):
$$\frac{125339771293231}{139403980800000}-\frac{4 \zeta (5)}{5}$$
For \(N=9\):
$$\frac{86351619628097899}{101625502003200000}-\frac{4 \zeta (5)}{5}$$
For \(N=10\):
$$\frac{3442878274477603}{4065020080128000}-\frac{4 \zeta (5)}{5}$$
For \(N=11\):
$$\frac{13123699834429625905481}{16366888723117363200000}-\frac{4 \zeta (5)}{5}$$
For \(N=12\):
$$\frac{13171485267700803325481}{16366888723117363200000}-\frac{4 \zeta (5)}{5}$$
For \(N=13\):
$$\frac{4629488048215941858586616933}{6076911214672415134617600000}-\frac{4 \zeta (5)}{5}$$
For \(N=14\):
$$\frac{4650213051562057979665016933}{6076911214672415134617600000}-\frac{4 \zeta (5)}{5}$$
For \(N=15\):
$$\frac{4416566295025256372880622181}{6076911214672415134617600000}-\frac{4 \zeta (5)}{5}$$
For \(N=16\):
$$\frac{142071262625911096676548515199}{194461158869517284307763200000}-\frac{4 \zeta (5)}{5}$$
For \(N=17\):
$$\frac{191003126077388028607407115680150293}{276107037648996202745367733862400000}-\frac{4 \zeta (5)}{5}$$
For \(N=18\):
$$\frac{192899639061612464522396079632790293}{276107037648996202745367733862400000}-\frac{4 \zeta (5)}{5}$$
For \(N=19\):
$$\frac{150453476396412876605169355404203944595669}{227889453271880616207200766816318259200000}-\frac{4 \zeta (5)}{5}$$
For \(N=20\):
$$\frac{151816292136537131422307455398824987089109}{227889453271880616207200766816318259200000}-\frac{4 \zeta (5)}{5}$$
For \(N=21\):
$$\frac{1149533260731205883813450761664862026041}{1823115626175044929657606134530546073600}-\frac{4 \zeta (5)}{5}$$
For \(N=22\):
$$\frac{1159675787637808723881934167302899780921}{1823115626175044929657606134530546073600}-\frac{4 \zeta (5)}{5}$$
For \(N=23\):
$$\frac{639608766586352776327542914664088015699022173}{1066745227156578837062475058249339864272076800}-\frac{4 \zeta (5)}{5}$$
For \(N=24\):
$$\frac{648359797309712362044617197585428118865809213}{1066745227156578837062475058249339864272076800}-\frac{4 \zeta (5)}{5}$$
For \(N=25\):
$$\frac{237944935065135444576929387506251376669625956690333}{416697354358038608227529319628648384481280000000000}-\frac{4 \zeta (5)}{5}$$
For \(N=26\):
$$\frac{240669046571084215593192115299853575191405294711197}{416697354358038608227529319628648384481280000000000}-\frac{4 \zeta (5)}{5}$$
For \(N=27\):
$$\frac{165073236825223636372867799350840561222210871844462613}{303772371327010145397868874009284672286853120000000000}-\frac{4 \zeta (5)}{5}$$
For \(N=28\):
$$\frac{167043357536724092371644378969941320805846771844462613}{303772371327010145397868874009284672286853120000000000}-\frac{4 \zeta (5)}{5}$$
For \(N=29\):
$$\frac{3203948973932129803689130844540389738329476286238317453292337}{6230720370371632816767352757266665296691815085434880000000000}-\frac{4 \zeta (5)}{5}$$
For \(N=30\):
$$\frac{3256925140401784859651721760665888583771641057876420235372337}{6230720370371632816767352757266665296691815085434880000000000}-\frac{4 \zeta (5)}{5}$$