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Exercises 18.3 Exercises

Exercise Group.

We see in Subsection 18.2.2 that \(r\) is not multiplicative. But could some related properties still be true?


Look at the cases where zero is involved. State the broadest possible multiplicativity result you can for this case.


Look at the second two examples in Subsection 18.2.2. There seems to be a specific sort of relationship in the precise way in which these examples are not multiplicative. What is that relationship? Can you prove it? (Hint: first compare the results, only then the individual inputs.)


For a fixed \(p(x)\text{,}\) let \(Z_{p(x)}(n)\) be the number of solutions of the polynomial congruence \(p(x)\equiv 0\text{ (mod }n)\text{.}\) Use facts from earlier in the text to show that this function is multiplicative. Connect this to the question of whether \(-1\in Q_n\text{.}\)


Let the function \(g\) be given by
\begin{equation*} g(n)=\begin{cases}\hfill 0 & n\text{ is even }\\\hfill 1 & n\equiv 1\text{ (mod }4)\\-1 & n\equiv 3\text{ (mod }4)\end{cases}\text{.} \end{equation*}
Show that the function \(g(n)\) is multiplicative.


Compute \(r(n)\) for \(0\leq n\leq 10\) and compare the sum to \(10\pi\text{.}\)


Compute \(r(n)\) for \(n=100\text{,}\) \(300\text{,}\) and \(900\text{.}\) Can you write down all the actual sums of squares for these?