Packing Characteristics of Different Shaped Proppants for use with Hydrofracing - A Numerical Investigation using 3D FEMDEM

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Note on sums involving the Euler function

In this note, we provide refined estimates of the following sums involving the Euler totient function: $$\sum_{n\le x} \phi\left(\left[\frac{x}{n}\right]\right) \qquad \text{and} \qquad \sum_{n\le x} \frac{\phi([x/n])}{[x/n]}$$ where $[x]$ denotes the integral part of real $x$. The above summations were recently considered by Bordell\`es et al. and Wu.Comment: Bull. Aust. Math. Soc., accepte