Competition problem — Number theory

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Number theory Difficulty 5.6 AIME, harder Prove it

5. When $\mu(k) \neq 0$ , the smallest positive period of $\chi(n ; k)$ (see the previous question) is $k$ .

Solution

5. When $\mu(k) \neq 0$ , for any $k>q \mid k$ , the character modulo $k$ is certainly not a character modulo $q$ . Then use the result from the previous question (ii).

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.