Number Theory: In Context and Interactive

Section 20.5 Looking Ahead

• 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 $\frac{\pi^2}{12}\text{.}$

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def L(n):

    ls = []

    out = 0

    for i in range(1,n+1):

        out += euler_phi(i)

        ls.append((i,out/i))

    return ls

​

LS = L(1000)

P = line(LS)

@interact

def _(a=.01,n=2,view=(50,[25,50,..500])):

    show(P+plot(a*x^n,0,view, color='black',linestyle="--"), xmin=1,xmax=view, ymax=LS[view][1])

    pretty_print(html(r"Blue is the average value of $\phi$"))

    pretty_print(html("Red is $%s x^{%s}$"%(latex(a),n)))