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Math Help - Sum of divisors of n

  1. #1
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    Sum of divisors of n


    Show that ∑ σ(n)*(1/ns) = ς(s) ς(s-1)
    n=1



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  2. #2
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    Quote Originally Posted by mndi1105 View Post

    Show that \sum_{n=1}^{\infty}\frac{\sigma(n)}{n^s} = \zeta(s) \zeta(s-1).
    \sum_{n=1}^{\infty}\frac{\sigma(n)}{n^s}=\sum_{n=1  }^{\infty}\frac{1}{n^s} \sum_{d \mid n} d=\sum_{d=1}^{\infty}d \sum_{n=1}^{\infty} \frac{1}{(dn)^s}=\sum_{d=1}^{\infty} \frac{1}{d^{s-1}} \sum_{n=1}^{\infty} \frac{1}{n^s}=\zeta(s-1) \zeta(s). \ \ \ \Box
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