Averages

These equivalent statements have the following shape :

$$\sum_{n \leq x} f(n) = F(x) + O(x^{\alpha + \epsilon}), \quad x \rightarrow +\infty,$$

where $f$ is an arithmetic function, $F(x)$ a smooth approximation to $\sum_{n \leq x} f(n)$, and $\alpha$ a real number.

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