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→Properties: proof of pseudo-associativity |
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<math>f*f = \tau \cdot f.</math>
Here <math> \tau</math> is the [[divisor function]].
===Proof of pseudo-associative property ===
<math> (f * g)(n) = \sum_{d|n} f(d) g \left( \frac{n}{d} \right)</math>
<math> f \cdot \left(g*h \right)(n) = f(n) \sum_{d|n} g(d) h \left( \frac{n}{d} \right) </math>
<math> f \cdot \left(g*h \right)(n) = \sum_{d|n} f(n) g(d) h \left( \frac{n}{d} \right) </math>
<math> f \cdot \left(g*h \right)(n) = \sum_{d|n} f(d) f \left( \frac{n}{d} \right) g(d) h \left( \frac{n}{d} \right) </math> (since ''f'' is completely multiplicative)
<math> f \cdot \left(g*h \right)(n) = \sum_{d|n} f(d) g(d) \left( \frac{n}{d} \right) h \left( \frac{n}{d} \right) </math>
<math> f \cdot \left(g*h \right)(n) = (f \cdot g)*(f \cdot h).</math>
==See also==
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