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Math Help - Zeta Function Proof

  1. #1
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    Zeta Function Proof

    Prove that sum(n=0 to infty, (zeta(it))^(n)) equals zero when the variable (it) is the imaginary part of the nontrivial zeros of the Riemann zeta function that have real part 1/2. For example, it=14.134i. Note: n represents the nth derivative of the zeta function.
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  2. #2
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    I tried summing the first 100 of them and initially looks like the sum is blowing up. However that could be due to some numerical problem with Mathematica computing the high-ordered derivatives correctly. I computed this sum:

    \sum_{n=0}^{25}\left( \frac{d^n \zeta(z)}{dz^n}\biggr|_{z=i \gamma_1}\right)

    where \gamma_1 is the imaginary part of the first non-trivial zero. I used the following Mathematica code and plotted the real part of partial sums below.

    Code:
    mynum = Im[ZetaZero[1]]; 
    mytable = Table[
       N[Re[Sum[D[Zeta[z], {z, n}], 
           {n, 0, nmax}] /. z -> I*mynum]], 
       {nmax, 1, 25}]
    That doesn't mean it's not zero of course. Only that the first numerical calculations do not suggest it is tending to zero.
    Attached Thumbnails Attached Thumbnails Zeta Function Proof-zetaderiv.jpg  
    Last edited by shawsend; January 26th 2010 at 07:13 AM.
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  3. #3
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    Infinite Derivative of the Zeta Function

    Thank you for your help. I was also wondering if f^{(\infty)}(z) equaled zero, where f(z) is the Riemann Zeta function and ^{(\infty)} is the infinite derivative of f(z).
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  4. #4
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    It's not hard to numerically check that:

    Code:
    mynum = Im[ZetaZero[1]];
    mytable = 
      Table[N[Re[D[Zeta[z], {z, n}] /. z -> I mynum]], {n, 1, 50}]
    and that list of values of \text{Re}\,\left(\frac{d^n\zeta(z)}{dz^n}\right)\b  iggr|_{z=i\gamma_1} is not tending to zero at least in the first 50 and Mathematica starts choking at around 100.

    Also, I am suspicious that Mathematica is even calculating these high-ordered derivatives correctly. You know that the derivatives of some functions start "blowing-up" in size as the order grows so if it were mine, I'd look for a second source to confirm at least the ones 0-100 are being computed correctly.
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  5. #5
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by PaulDirac2 View Post
    Prove that sum(n=0 to infty, (zeta(it))^(n)) equals zero when the variable (it) is the imaginary part of the nontrivial zeros of the Riemann zeta function that have real part 1/2. For example, it=14.134i. Note: n represents the nth derivative of the zeta function.
    Where did you find this problem?
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  6. #6
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    Problem

    My teacher assigned it to our class. It was extra credit.
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  7. #7
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by PaulDirac2 View Post
    My teacher assigned it to our class. It was extra credit.
    Let me know what the solution is. This is very interesting.
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