Atal's Integrals

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This page contains the values of the following integral as $N$ varies: $$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_{\frac{11}{2}}(t).$$ Here $\zeta$ is the Riemann zeta function.

For $N=1$: $$\frac{9}{11}-\frac{11 \zeta \left(\frac{13}{2}\right)}{13}$$ For $N=2$: $$-\frac{11 \zeta \left(\frac{13}{2}\right)}{13}+\frac{29}{22}+\frac{1}{64 \sqrt{2}}$$ For $N=3$: $$-\frac{11 \zeta \left(\frac{13}{2}\right)}{13}+\frac{38}{33}+\frac{16 \sqrt{\frac{2}{3}}}{729}+\frac{1}{64 \sqrt{2}}+\frac{1}{729 \sqrt{3}}$$ For $N=4$: $$-\frac{11 \zeta \left(\frac{13}{2}\right)}{13}+\frac{96267}{90112}+\frac{3}{128 \sqrt{2}}+\frac{185339}{5971968 \sqrt{3}}$$ For $N=5$: $$-\frac{11 \zeta \left(\frac{13}{2}\right)}{13}+\frac{5800651}{5857280}+\frac{16 \sqrt{\frac{2}{5}}}{15625}+\frac{162 \sqrt{\frac{3}{5}}}{15625}+\frac{63}{2860 \sqrt{2}}+\frac{1}{729 \sqrt{3}}+\frac{513}{15625 \sqrt{5}}$$ For $N=6$: $$-\frac{11 \zeta \left(\frac{13}{2}\right)}{13}+\frac{3363245}{3514368}+\frac{625 \sqrt{\frac{5}{6}}}{46656}+\frac{805}{27456 \sqrt{2}}+\frac{1}{486 \sqrt{3}}+\frac{1}{15625 \sqrt{5}}+\frac{1025}{46656 \sqrt{6}}$$ For $N=7$: $$-\frac{11 \zeta \left(\frac{13}{2}\right)}{13}+\frac{7422953}{8200192}+\frac{16 \sqrt{\frac{2}{7}}}{117649}+\frac{81 \sqrt{\frac{3}{7}}}{117649}+\frac{1250 \sqrt{\frac{5}{7}}}{117649}+\frac{1296 \sqrt{\frac{6}{7}}}{117649}+\frac{1993}{64064 \sqrt{2}}+\frac{1}{729 \sqrt{3}}+\frac{1}{15625 \sqrt{5}}+\frac{1}{46656 \sqrt{6}}+\frac{1537}{117649 \sqrt{7}}$$ For $N=8$: $$-\frac{11 \zeta \left(\frac{13}{2}\right)}{13}+\frac{2079149}{2342912}+\frac{2834575}{74973184 \sqrt{2}}+\frac{1875 \sqrt{\frac{5}{2}}}{524288}+\frac{193531}{11943936 \sqrt{3}}+\frac{2401 \sqrt{\frac{7}{2}}}{524288}+\frac{1}{15625 \sqrt{5}}+\frac{181243}{191102976 \sqrt{6}}+\frac{1}{117649 \sqrt{7}}$$ For $N=9$: $$-\frac{11 \zeta \left(\frac{13}{2}\right)}{13}+\frac{58508212327}{69173305344}+\frac{227474397287}{4427091542016 \sqrt{2}}+\frac{4}{2187 \sqrt{3}}+\frac{196906823}{24911296875 \sqrt{5}}+\frac{2051}{139968 \sqrt{6}}+\frac{3956247809}{187570506627 \sqrt{7}}$$ For $N=10$: $$-\frac{11 \zeta \left(\frac{13}{2}\right)}{13}+\frac{7791849778769}{9338396221440}+\frac{81 \sqrt{\frac{3}{10}}}{1000000}+\frac{81 \sqrt{\frac{3}{5}}}{15625}+\frac{7203 \sqrt{\frac{7}{10}}}{1000000}+\frac{17744587}{374865920 \sqrt{2}}+\frac{1}{729 \sqrt{3}}+\frac{103}{6250 \sqrt{5}}+\frac{1}{46656 \sqrt{6}}+\frac{1}{117649 \sqrt{7}}+\frac{5177}{250000 \sqrt{10}}$$