# MAT 338 Day 36 2011

## 3263 days ago by kcrisman

Recall that last time we were finishing up discussing long-term average values of certain arithmetic functions.  We had that

• The average value of $\tau(n)$ was $\ln(n)+2\gamma-1$.
• 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 $$\sum_{d=1}^\infty \frac{1}{d^2}=\frac{\pi^2}{6}$$ so the constant in question is $\frac{\pi^2}{12}$.

We will discuss this computation again soon, when we return to the connection between number theory and such abstract series.

We ended with the question of yet another average value - that of the $\phi$ function.  You can try out various ideas below.  However, we aren't ready to prove anything about that quite yet.

def L(n): ls = [] out = 0 for i in range(1,n+1): out += euler_phi(i) ls.append((i,out/i)) return ls P = line(L(100))