Math Help - Zeta Function Proof

1. 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.

2. 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.

3. 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).

4. 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.

5. Originally Posted by PaulDirac2
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?

6. Problem

My teacher assigned it to our class. It was extra credit.

7. Originally Posted by PaulDirac2
My teacher assigned it to our class. It was extra credit.
Let me know what the solution is. This is very interesting.