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Thread: zeta function

  1. #1
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    zeta function

    why is this correct?
    If $\displaystyle \zeta(s)= \Sigma ^\infty _{n=1} {1 \over {n^s}}$ then $\displaystyle \zeta(s)= \Sigma ^\infty _{N=1} {\tau(N) \over {N^s}}$ where $\displaystyle \tau(N)$ the number of divisors of N...

    Also
    $\displaystyle \zeta(s)\zeta(s-1) = \Sigma {\sigma(N) \over N^s}$ where $\displaystyle \sigma(N)$ is the sum of the divisors of N....
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  2. #2
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    Quote Originally Posted by bigdoggy View Post
    why is this correct?
    If $\displaystyle \zeta(s)= \Sigma ^\infty _{n=1} {1 \over {n^s}}$ then $\displaystyle \zeta(s)= \Sigma ^\infty _{N=1} {\tau(N) \over {N^s}}$ where $\displaystyle \tau(N)$ the number of divisors of N...

    Also
    $\displaystyle \zeta(s)\zeta(s-1) = \Sigma {\sigma(N) \over N^s}$ where $\displaystyle \sigma(N)$ is the sum of the divisors of N....
    the LHS of the first identity is wrong. it should be $\displaystyle \zeta^2(s).$
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  3. #3
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    ok, for the integers $\displaystyle k \geq 0, \ n \geq 1,$ let $\displaystyle f_k(n)=\sum_{d \mid n} d^k.$ then, assuming that $\displaystyle \zeta(s-k)$ is defined we have: $\displaystyle \sum_{n=1}^{\infty} \frac{f_k(n)}{n^s}=\sum_{n=1}^{\infty} \sum_{d \mid n} \frac{d^k}{n^s}=\sum_{d=1}^{\infty} d^k \sum_{m=1}^{\infty} \frac{1}{(md)^s}=\sum_{d=1}^{\infty} \frac{1}{d^{s-k}} \sum_{m=1}^{\infty} \frac{1}{m^s}=\zeta(s-k) \zeta(s).$
    now use the fact that $\displaystyle f_0(n)=\tau(n)$ and $\displaystyle f_1(n)=\sigma(n).$
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