Atal's Integrals
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This page contains the values of the following integral as \(N\) varies:
$$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_{\frac{9}{2}}(t).$$
Here \(\zeta\) is the Riemann zeta function.
For \(N=1\):
$$\frac{7}{9}-\frac{9 \zeta \left(\frac{11}{2}\right)}{11}$$
For \(N=2\):
$$-\frac{9 \zeta \left(\frac{11}{2}\right)}{11}+\frac{23}{18}+\frac{1}{32 \sqrt{2}}$$
For \(N=3\):
$$-\frac{9 \zeta \left(\frac{11}{2}\right)}{11}+\frac{10}{9}+\frac{1}{32 \sqrt{2}}+\frac{1}{729} \left(\sqrt{3}+8 \sqrt{6}\right)$$
For \(N=4\):
$$\frac{1535193+34992 \sqrt{2}+21731 \sqrt{3}}{1492992}-\frac{9 \zeta \left(\frac{11}{2}\right)}{11}$$
For \(N=5\):
$$-\frac{9 \zeta \left(\frac{11}{2}\right)}{11}+\frac{950767}{1013760}+\frac{8 \sqrt{\frac{2}{5}}}{3125}+\frac{54 \sqrt{\frac{3}{5}}}{3125}+\frac{181}{3960 \sqrt{2}}+\frac{1}{243 \sqrt{3}}+\frac{129}{3125 \sqrt{5}}$$
For \(N=6\):
$$-\frac{9 \zeta \left(\frac{11}{2}\right)}{11}+\frac{183395}{202752}+\frac{125 \sqrt{\frac{5}{6}}}{7776}+\frac{191}{3168 \sqrt{2}}+\frac{1}{162 \sqrt{3}}+\frac{1}{3125 \sqrt{5}}+\frac{257}{7776 \sqrt{6}}$$
For \(N=7\):
$$-\frac{9 \zeta \left(\frac{11}{2}\right)}{11}+\frac{397543}{473088}+\frac{8 \sqrt{\frac{2}{7}}}{16807}+\frac{27 \sqrt{\frac{3}{7}}}{16807}+\frac{250 \sqrt{\frac{5}{7}}}{16807}+\frac{216 \sqrt{\frac{6}{7}}}{16807}+\frac{1453}{22176 \sqrt{2}}+\frac{1}{243 \sqrt{3}}+\frac{1}{3125 \sqrt{5}}+\frac{1}{7776 \sqrt{6}}+\frac{55}{2401 \sqrt{7}}$$
For \(N=8\):
$$-\frac{9 \zeta \left(\frac{11}{2}\right)}{11}+\frac{111203}{135168}+\frac{512099}{6488064 \sqrt{2}}+\frac{375 \sqrt{\frac{5}{2}}}{65536}+\frac{23779}{995328 \sqrt{3}}+\frac{343 \sqrt{\frac{7}{2}}}{65536}+\frac{1}{3125 \sqrt{5}}+\frac{20707}{7962624 \sqrt{6}}+\frac{1}{16807 \sqrt{7}}$$
For \(N=9\):
$$-\frac{9 \zeta \left(\frac{11}{2}\right)}{11}+\frac{1022824771}{1330255872}+\frac{4205218115}{42568187904 \sqrt{2}}+\frac{4}{729 \sqrt{3}}+\frac{7989647}{553584375 \sqrt{5}}+\frac{515}{23328 \sqrt{6}}+\frac{80884361}{2977309629 \sqrt{7}}$$
For \(N=10\):
$$-\frac{9 \zeta \left(\frac{11}{2}\right)}{11}+\frac{15106244333}{19953838080}+\frac{27 \sqrt{\frac{3}{10}}}{100000}+\frac{27 \sqrt{\frac{3}{5}}}{3125}+\frac{1029 \sqrt{\frac{7}{10}}}{100000}+\frac{1077413}{10813440 \sqrt{2}}+\frac{1}{243 \sqrt{3}}+\frac{131}{6250 \sqrt{5}}+\frac{1}{7776 \sqrt{6}}+\frac{1}{16807 \sqrt{7}}+\frac{611}{25000 \sqrt{10}}$$