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Math Help - More on Sum of the divisors

  1. #1
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    More on Sum of the divisors

    Let σ2(n) = ∑d|nd2. Show that ∑ σ2(n)/n4 = π 6/540
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  2. #2
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    Quote Originally Posted by mndi1105 View Post

    Let \sigma_2(n) = \sum_{d \mid n}d^2. Show that \sum_{n=1}^{\infty} \frac{\sigma_2(n)}{n^4} = \frac{\pi^6}{540}.
    \sum_{n=1}^{\infty} \frac{\sigma_2(n)}{n^4}=\sum_{n=1}^{\infty}\frac{1  }{n^4} \sum_{d \mid n} d^2=\sum_{d=1}^{\infty}d^2 \sum_{n=1}^{\infty} \frac{1}{(dn)^4}=\sum_{d=1}^{\infty} \frac{1}{d^2} \sum_{n=1}^{\infty} \frac{1}{n^4}=\frac{\pi^2}{6} \times \frac{\pi^4}{90}=\frac{\pi^6}{540}. \ \ \Box
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