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I am trying exercises of Apostol Introduction to analytic number theory and I am struck on this problem of chapter partitions on page 14.

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I am struck in part (b) as I have no idea on how to deal with the problem as I can put x=x times ${\alpha}^h$ and then product in h=1 to 4 but how to prove that there will be powers of $ \alpha $ along with I's.

Can you please help proving it?

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    $\begingroup$ Can you show that if $S(x)$ is a power series of type $r$ mod $q$ and $\alpha^q=1$ then $$S(\alpha x)=\alpha^r S(x)?$$ $\endgroup$
    – Jyrki Lahtonen
    Commented Oct 26, 2020 at 8:26
  • $\begingroup$ What is confusing you after @JyrkiLahtonen's comment? $\endgroup$
    – JMP
    Commented May 15, 2021 at 11:44
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    $\begingroup$ Adding to @JyrkiLahtonen's point, $\prod_{n=1}^\infty(1-x^n\alpha^{nh})=\prod_{n=1}^\infty(1-(x\alpha^h)^n)=\varphi(x\alpha^h)$. $\endgroup$
    – Kenta S
    Commented May 17, 2021 at 14:49

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