對於正整數 $k\geqslant 3$, 正整數 $a_1,\dotsc, a_k \in \mathbb{N}_{+}$ 互素 $(a_1,\dotsc,a_k)=1$.
\begin{align*}
& \quad \sum_{n_1 \leqslant x} \dotsc \sum_{n_{k} \leqslant x}
\left\{ \frac{x}{a_1n_1+\dotsb+a_kn_k} \right\} \\
& = \Bigg( \frac{1}{(k-1)! a_1 \dotsm a_k} \bigg(
\sum_{1\leqslant i \leqslant k} (-1)^{k+1} a_{i}^{k-1} \log a_{i}
+ \sum_{1\leqslant i_1 < i_2 \leqslant k} (-1)^{k+2} (a_{i_{1}}+a_{i_{2}})^{k-1} \log(a_{i_1}+a_{i_2}) \\
& \quad + \sum_{1\leqslant i_1 < i_2 <i_3 \leqslant k} (-1)^{k+3} (a_{i_{1}}+a_{i_{2}}+a_{i_{3}})^{k-1} \log(a_{i_1}+a_{i_2}+a_{i_3}) + \dotsb \\
&\quad + (a_1+\dotsb+a_k)^{k-1} \log(a_1+\dotsb +a_k) \bigg) - \frac{\zeta(k)}{k! a_1\dotsm a_k} \Bigg)x^{k} + O(x^{k-1})
\end{align*}

對於正整數 $k\geqslant 2$, 正整數 $a_1,\dotsc, a_k \in \mathbb{N}_{+}$ 互素 $(a_1,\dotsc,a_k)=1$.
\begin{align*}
&\quad \int_{0}^{1} \dotsi \int_{0}^{1} \left\{ \frac{1}{a_1t_1+\dotsb+a_kt_k} \right\} \, \mathrm{d}t_1 \dotsm \mathrm{d}t_k \\
& = \frac{1}{(k-1)! a_1 \dotsm a_k} \bigg(
\sum_{1\leqslant i \leqslant k} (-1)^{k+1} a_{i}^{k-1} \log a_{i}
+ \sum_{1\leqslant i_1 < i_2 \leqslant k} (-1)^{k+2} (a_{i_{1}}+a_{i_{2}})^{k-1} \log(a_{i_1}+a_{i_2}) \\
& \quad + \sum_{1\leqslant i_1 < i_2 <i_3 \leqslant k} (-1)^{k+3} (a_{i_{1}}+a_{i_{2}}+a_{i_{3}})^{k-1} \log(a_{i_1}+a_{i_2}+a_{i_3}) + \dotsb \\
&\quad + (a_1+\dotsb+a_k)^{k-1} \log(a_1+\dotsb +a_k) \bigg) - \frac{\zeta(k)}{k! a_1\dotsm a_k}
\end{align*}
當 $k=1$ 時, 實數 $a_1 \geqslant 1$
\begin{equation*}
\int_{0}^{1} \left\{ \frac{1}{a_{1}t_{1}} \right\} \, \mathrm{d}t_1 = \frac{1- \gamma +\log a_{1}}{a_{1}}.
\end{equation*}