Atal's Integrals

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Given a number \(r\), define a function \(f_r\) on \([0,1]\) as follows: $$f_r(x) =\begin{cases} x^r \left( \frac{1}{x}-n \right) & \text{if } n \in \mathbb{N} \text{ and } \frac{1}{n+1} \le x \leq \frac{1}{n}, \\ 0 & \text{if } x = 0. \end{cases}$$ If \(r > 1\), \(f_r\) has bounded variation and so we can consider Stieltjes integrals with respect to \(f_r\).
The following links contain pages with the values of the sum $$\sum_{m=0}^{N-1} \int_{m/N}^{(m+1)/N} \left( t - \frac{m}{N} \right) \,df_r(t)$$ as \(N\) and \(r\) vary.

For integer \(r\):


For half-integer \(r\):