Let's recap.

• The average value of \(\tau(n)\) was \(\log(n)+2\gamma-1\).

• The average value of \(\sigma(n)\) was \(\left(\frac{1}{2}\sum_{d=1}^\infty \frac{1}{d^2}\right)\; n\).

  • 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 \(\frac{\pi^2}{12}\).

We end with the question of yet another average value. What might happen with the \(\phi\) function? You can try out various ideas below; a is the coefficient and n is the power of a model \(ax^n\).

Hopefully you started finding something interesting. However, we aren't ready to prove anything about that quite yet.