Atal's Integrals
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This page contains the values of the following integral as \(N\) varies:
$$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_{\frac{7}{2}}(t).$$
Here \(\zeta\) is the Riemann zeta function.
For \(N=1\):
$$\frac{5}{7}-\frac{7 \zeta \left(\frac{9}{2}\right)}{9}$$
For \(N=2\):
$$-\frac{7 \zeta \left(\frac{9}{2}\right)}{9}+\frac{17}{14}+\frac{1}{16 \sqrt{2}}$$
For \(N=3\):
$$-\frac{7 \zeta \left(\frac{9}{2}\right)}{9}+\frac{22}{21}+\frac{1}{16 \sqrt{2}}+\frac{1}{81} \sqrt{11+\frac{8 \sqrt{2}}{3}}$$
For \(N=4\):
$$-\frac{7 \zeta \left(\frac{9}{2}\right)}{9}+\frac{3463}{3584}+\frac{3}{32 \sqrt{2}}+\frac{2699}{41472 \sqrt{3}}$$
For \(N=5\):
$$-\frac{7 \zeta \left(\frac{9}{2}\right)}{9}+\frac{137531}{161280}+\frac{4 \sqrt{\frac{2}{5}}}{625}+\frac{18 \sqrt{\frac{3}{5}}}{625}+\frac{61}{630 \sqrt{2}}+\frac{1}{81 \sqrt{3}}+\frac{33}{625 \sqrt{5}}$$
For \(N=6\):
$$-\frac{7 \zeta \left(\frac{9}{2}\right)}{9}+\frac{26431}{32256}+\frac{25 \sqrt{\frac{5}{6}}}{1296}+\frac{127}{1008 \sqrt{2}}+\frac{1}{54 \sqrt{3}}+\frac{1}{625 \sqrt{5}}+\frac{65}{1296 \sqrt{6}}$$
For \(N=7\):
$$-\frac{7 \zeta \left(\frac{9}{2}\right)}{9}+\frac{23615}{32256}+\frac{4 \sqrt{\frac{2}{7}}}{2401}+\frac{9 \sqrt{\frac{3}{7}}}{2401}+\frac{50 \sqrt{\frac{5}{7}}}{2401}+\frac{36 \sqrt{\frac{6}{7}}}{2401}+\frac{143}{1008 \sqrt{2}}+\frac{1}{81 \sqrt{3}}+\frac{1}{625 \sqrt{5}}+\frac{1}{1296 \sqrt{6}}+\frac{97}{2401 \sqrt{7}}$$
For \(N=8\):
$$-\frac{7 \zeta \left(\frac{9}{2}\right)}{9}+\frac{46141}{64512}+\frac{87103}{516096 \sqrt{2}}+\frac{75 \sqrt{\frac{5}{2}}}{8192}+\frac{3211}{82944 \sqrt{3}}+\frac{49 \sqrt{\frac{7}{2}}}{8192}+\frac{1}{625 \sqrt{5}}+\frac{2443}{331776 \sqrt{6}}+\frac{1}{2401 \sqrt{7}}$$
For \(N=9\):
$$-\frac{7 \zeta \left(\frac{9}{2}\right)}{9}+\frac{15015271}{23514624}+\frac{76755815}{376233984 \sqrt{2}}+\frac{4}{243 \sqrt{3}}+\frac{332183}{12301875 \sqrt{5}}+\frac{131}{3888 \sqrt{6}}+\frac{1666769}{47258883 \sqrt{7}}$$
For \(N=10\):
$$-\frac{7 \zeta \left(\frac{9}{2}\right)}{9}+\frac{220736521}{352719360}+\frac{9 \sqrt{\frac{3}{10}}}{10000}+\frac{9 \sqrt{\frac{3}{5}}}{625}+\frac{147 \sqrt{\frac{7}{10}}}{10000}+\frac{185449}{860160 \sqrt{2}}+\frac{1}{81 \sqrt{3}}+\frac{7}{250 \sqrt{5}}+\frac{1}{1296 \sqrt{6}}+\frac{1}{2401 \sqrt{7}}+\frac{77}{2500 \sqrt{10}}$$