In this note, we provide refined estimates of two sums involving the Euler totient function, $$\begin{eqnarray}\mathop{\sum }_{n\leq x}\unicode[STIX]{x1D719}\biggl(\biggl[\frac{x}{n}\biggr]\biggr)\quad \text{and}\quad \mathop{\sum }_{n\leq x}\frac{\unicode[STIX]{x1D719}([x/n])}{[x/n]},\end{eqnarray}$$ where $[x]$ denotes the integral part of real $x$ . The above summations were recently considered by Bordellès et al. [‘On a sum involving the Euler function’, Preprint, 2018, arXiv:1808.00188] and Wu [‘On a sum involving the Euler totient function’, Preprint, 2018, hal-01884018].

在本说明中，我们提供了涉及欧拉totient函数的两个和的精确估计： $$ \ begin {eqnarray} \ mathop {\ sum __ {n \ leq x} \ unicode [STIX] {x1D719} \ biggl（\ biggl [\ frac {x} {n} \ biggr] \ biggr）\ quad \ text {and} \ quad \ mathop {\ sum} _ {n \ leq x} \ frac {\ unicode [STIX] {x1D719}（[ x / n]）} {[x / n]}，\ end {eqnarray} $$ 其中 $ [x] $ 表示实数 $ x $ 的整数部分。Bordellès等人最近考虑了上述总结。[“关于涉及欧拉函数的总和”，预印本，2018，arXiv：1808.00188]和Wu [“关于涉及欧拉函数的总和”，预印本，2018，hal-01884018]。