Number Theory: In Context and Interactive

6.

Find a formula for the average value of the $u$ and $N$ functions (up through $n$), where $u(n)=1$ for all $n$ and $N(n)=n$ for all $n$ (recall Definition 19.2.9).

7.

As suggested at the end of Subsection 20.2.1, if you are familiar with semilog and log-log plots and how to use them to find possible formulas, look up how to use them in Sage and modify the examples to explore whether the average value of $\tau$ could be a power function, exponential, or something else.

8.

At the start of Subsection 20.2.1 we plot the cumulative average value of $\tau \text{.}$ Note that because

this value is the same for $n=4,5\text{.}$ Is there ever an $n>5$ where this happens again?

9.

Finish off all calculus details in the argument in Subsection 20.2.3.

10.

Finish the details of the proof that $\tau$ is $O(\sqrt[3]{x})$

11.

Show that $\tau (n)$ is not $O(1)\text{.}$ (Hint: that means there is no constant $C$ such that $\tau (n)\le C$ always.)

12.

Suppose that for an arithmetic function $f$ it is known that $\frac{1}{n}\sum _{k=1}^{n}f(k)=O(1)\text{;}$ why is it still possible that $f(n)$ is not $O(1)\text{?}$

13.

Show that $\tau (n)$ is not $O(\log (n))\text{,}$ even though it is known that $\frac{1}{n}\sum _{k=1}^n\tau (k)=O(\log (n))\text{.}$ (Hint: look at numbers of the form $6^k\text{,}$ and compare $\tau$ of these to any given multiple of the natural logarithm using calculus.)

14.

Finish all calculus details of the proof of $\sigma$'s average size in Section 20.4.

15.

Finish the details of the first computation of Big Oh in Subsection 20.4.2.

16.

Find absolute bounds for $\varphi (n)$ (simple polynomial or log formulas in terms of $n$).

17.

Use data, graphs, whatever to conjecture what type of growth the average value of $\varphi$ has up to $n\text{.}$ Is it logarithmic, linear, quadratic, exponential, something else? Bonus if you find a coefficient for the growth!