Atal's Integrals
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This page contains the values of the following integral as \(N\) varies:
$$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_{\frac{5}{2}}(t).$$
Here \(\zeta\) is the Riemann zeta function.
For \(N=1\):
$$\frac{3}{5}-\frac{5 \zeta \left(\frac{7}{2}\right)}{7}$$
For \(N=2\):
$$\frac{1}{80} \left(88+5 \sqrt{2}\right)-\frac{5 \zeta \left(\frac{7}{2}\right)}{7}$$
For \(N=3\):
$$-\frac{5 \zeta \left(\frac{7}{2}\right)}{7}+\frac{14}{15}+\frac{1}{8 \sqrt{2}}+\frac{1}{27} \sqrt{3+\frac{4 \sqrt{2}}{3}}$$
For \(N=4\):
$$-\frac{5 \zeta \left(\frac{7}{2}\right)}{7}+\frac{549}{640}+\frac{3}{16 \sqrt{2}}+\frac{371}{3456 \sqrt{3}}$$
For \(N=5\):
$$-\frac{5 \zeta \left(\frac{7}{2}\right)}{7}+\frac{3107}{4480}+\frac{2 \sqrt{\frac{2}{5}}}{125}+\frac{6 \sqrt{\frac{3}{5}}}{125}+\frac{3}{14 \sqrt{2}}+\frac{1}{27 \sqrt{3}}+\frac{9}{125 \sqrt{5}}$$
For \(N=6\):
$$-\frac{5 \zeta \left(\frac{7}{2}\right)}{7}+\frac{8873}{13440}+\frac{5 \sqrt{\frac{5}{6}}}{216}+\frac{229}{840 \sqrt{2}}+\frac{1}{18 \sqrt{3}}+\frac{1}{125 \sqrt{5}}+\frac{17}{216 \sqrt{6}}$$
For \(N=7\):
$$-\frac{5 \zeta \left(\frac{7}{2}\right)}{7}+\frac{2339}{4480}+\frac{2 \sqrt{\frac{2}{7}}}{343}+\frac{3 \sqrt{\frac{3}{7}}}{343}+\frac{10 \sqrt{\frac{5}{7}}}{343}+\frac{6 \sqrt{\frac{6}{7}}}{343}+\frac{13}{40 \sqrt{2}}+\frac{1}{27 \sqrt{3}}+\frac{1}{125 \sqrt{5}}+\frac{1}{216 \sqrt{6}}+\frac{25}{343 \sqrt{7}}$$
For \(N=8\):
$$-\frac{5 \zeta \left(\frac{7}{2}\right)}{7}+\frac{4553}{8960}+\frac{13603}{35840 \sqrt{2}}+\frac{15 \sqrt{\frac{5}{2}}}{1024}+\frac{499}{6912 \sqrt{3}}+\frac{7 \sqrt{\frac{7}{2}}}{1024}+\frac{1}{125 \sqrt{5}}+\frac{307}{13824 \sqrt{6}}+\frac{1}{343 \sqrt{7}}$$
For \(N=9\):
$$-\frac{5 \zeta \left(\frac{7}{2}\right)}{7}+\frac{413881}{1088640}+\frac{4008761}{8709120 \sqrt{2}}+\frac{4}{81 \sqrt{3}}+\frac{14687}{273375 \sqrt{5}}+\frac{35}{648 \sqrt{6}}+\frac{35801}{750141 \sqrt{7}}$$
For \(N=10\):
$$-\frac{5 \zeta \left(\frac{7}{2}\right)}{7}+\frac{716045}{1959552}+\frac{3 \sqrt{\frac{3}{10}}}{1000}+\frac{3 \sqrt{\frac{3}{5}}}{125}+\frac{21 \sqrt{\frac{7}{10}}}{1000}+\frac{17699}{35840 \sqrt{2}}+\frac{1}{27 \sqrt{3}}+\frac{11}{250 \sqrt{5}}+\frac{1}{216 \sqrt{6}}+\frac{1}{343 \sqrt{7}}+\frac{11}{250 \sqrt{10}}$$