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== Summatory functions ==
Given an additive function <math>f</math>, let its summatory function be defined by <math display="inline">\mathcal{M}_f(x) := \sum_{n \leq x} f(n)</math>. The average of <math>f</math> is given exactly as
<math display=block>\mathcal{M}_f(x) = \sum_{p^{\alpha} \leq x} f(p^{\alpha}) \left(\left\lfloor \frac{x}{p^{\alpha}} \right\rfloor - \left\lfloor \frac{x}{p^{\alpha+1}} \right\rfloor\right).</math>
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