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Exercises 18.3 Exercises

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

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1.

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

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2.

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.)

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3.

For a fixed p(x), let Zp(x)(n) be the number of solutions of the polynomial congruence p(x)≡0 (mod n). Use facts from earlier in the text to show that this function is multiplicative. Connect this to the question of whether −1∈Qn.

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4.

Let the function g be given by

g(n)={0n is even 1n≡1 (mod 4)−1n≡3 (mod 4).

Show that the function g(n) is multiplicative.

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5.

Compute r(n) for 0≤n≤10 and compare the sum to 10π.

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6.

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