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Math Help - Euler's Function

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    Euler's Function

    Show that ∑ Φ(n)*(1/ns) = ς(s-1)/ς(s)
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  2. #2
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    Quote Originally Posted by mndi1105 View Post

    Show that \sum_{n=1}^{\infty}\frac{\varphi(n)}{n^s}=\frac{\z  eta(s-1)}{\zeta(s)}.
    \zeta(s)\sum_{n=1}^{\infty}\frac{\varphi(n)}{n^s}=  \sum_{m=1}^{\infty} \frac{1}{m^s}\sum_{n=1}^{\infty}\frac{\varphi(n)}{  n^s}=\sum_{k=1}^{\infty}\frac{1}{k^s}\sum_{d \mid k} \varphi(d)=\sum_{k=1}^{\infty} \frac{1}{k^{s-1}}=\zeta(s-1). \ \ \ \Box
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