Atal's Integrals
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This page contains the values of the following integral as \(N\) varies:
$$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_{\frac{3}{2}}(t).$$
Here \(\zeta\) is the Riemann zeta function.
For \(N=1\):
$$\frac{1}{3}-\frac{3 \zeta \left(\frac{5}{2}\right)}{5}$$
For \(N=2\):
$$-\frac{3 \zeta \left(\frac{5}{2}\right)}{5}+\frac{5}{6}+\frac{1}{4 \sqrt{2}}$$
For \(N=3\):
$$\frac{1}{216} \left(144+27 \sqrt{2}+8 \sqrt{3}+8 \sqrt{6}\right)-\frac{3 \zeta \left(\frac{5}{2}\right)}{5}$$
For \(N=4\):
$$\frac{1}{864} \left(531+162 \sqrt{2}+59 \sqrt{3}\right)-\frac{3 \zeta \left(\frac{5}{2}\right)}{5}$$
For \(N=5\):
$$-\frac{3 \zeta \left(\frac{5}{2}\right)}{5}+\frac{143}{480}+\frac{\sqrt{\frac{2}{5}}}{25}+\frac{2 \sqrt{\frac{3}{5}}}{25}+\frac{4 \sqrt{2}}{15}+\frac{1}{9 \sqrt{3}}+\frac{3}{25 \sqrt{5}}$$
For \(N=6\):
$$-\frac{3 \zeta \left(\frac{5}{2}\right)}{5}+\frac{127}{480}+\frac{\sqrt{\frac{5}{6}}}{36}+\frac{13}{20 \sqrt{2}}+\frac{1}{6 \sqrt{3}}+\frac{1}{25 \sqrt{5}}+\frac{5}{36 \sqrt{6}}$$
For \(N=7\):
$$-\frac{3 \zeta \left(\frac{5}{2}\right)}{5}-\frac{29}{1120}+\frac{\sqrt{\frac{2}{7}}}{49}+\frac{\sqrt{\frac{3}{7}}}{49}+\frac{2 \sqrt{\frac{5}{7}}}{49}+\frac{\sqrt{\frac{6}{7}}}{49}+\frac{361}{420 \sqrt{2}}+\frac{1}{9 \sqrt{3}}+\frac{1}{25 \sqrt{5}}+\frac{1}{36 \sqrt{6}}+\frac{1}{7 \sqrt{7}}$$
For \(N=8\):
$$-\frac{3 \zeta \left(\frac{5}{2}\right)}{5}-\frac{9}{320}+\frac{1871}{1920 \sqrt{2}}+\frac{3 \sqrt{\frac{5}{2}}}{128}+\frac{91}{576 \sqrt{3}}+\frac{\sqrt{\frac{7}{2}}}{128}+\frac{1}{25 \sqrt{5}}+\frac{43}{576 \sqrt{6}}+\frac{1}{49 \sqrt{7}}$$
For \(N=9\):
$$-\frac{3 \zeta \left(\frac{5}{2}\right)}{5}-\frac{4043}{12960}+\frac{63893}{51840 \sqrt{2}}+\frac{4}{27 \sqrt{3}}+\frac{743}{6075 \sqrt{5}}+\frac{11}{108 \sqrt{6}}+\frac{929}{11907 \sqrt{7}}$$
For \(N=10\):
$$-\frac{3 \zeta \left(\frac{5}{2}\right)}{5}-\frac{12881}{38880}+\frac{\sqrt{\frac{3}{10}}}{100}+\frac{\sqrt{\frac{2}{5}}}{25}+\frac{\sqrt{\frac{3}{5}}}{25}+\frac{3 \sqrt{\frac{7}{10}}}{100}+\frac{837}{640 \sqrt{2}}+\frac{1}{9 \sqrt{3}}+\frac{1}{10 \sqrt{5}}+\frac{1}{36 \sqrt{6}}+\frac{1}{49 \sqrt{7}}$$