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Section 20.5 Looking Ahead

Let’s recap.
  • The average value of \(\tau(n)\) was \(\log(n)+2\gamma-1\text{.}\)
  • The average value of \(\sigma(n)\) was \(\left(\frac{1}{2}\sum_{d=1}^\infty \frac{1}{d^2}\right)\; n\text{.}\)
    • Because of Euler’s amazing solution to the Basel problem, we know that
      \begin{equation*} \sum_{d=1}^\infty \frac{1}{d^2}=\frac{\pi^2}{6} \end{equation*}
      so the constant in question is \(\frac{\pi^2}{12}\text{.}\)
We end with the question of yet another average value. What might happen with the \(\phi\) function? You can try out various ideas in the following interact. Note that a is the coefficient and n is the power of a model \(ax^n\text{.}\)
Hopefully you started finding something interesting. However, we aren’t ready to prove anything about this case quite – yet!