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I'm trying a computational experiment where I am computing $\zeta(s)$ for $s= \frac{1}{2}+ iw$ by separately computing

  1. The contribution of all the non-trivial zeros (locations obtained from tables found on the web);

  2. The pole at $s=1,$. and

  3. The contribution of all the trivial zeros at negative even integers, $$\prod_{n=1}^{a\ very\ large\ number}(1 + \frac{s}{2n})$$ (a very large number limited by computing power).

Unfortunately, this last expression for the contribution of the trivial zeros does not converge for $s=\frac{1}{2} + iw.$

Are there any tricks to circumvent this problem?

Glorfindel
  • 3,955
Bob
  • 161
  • The zeta-function has a Hadamard product representation. Look that up. It includes an exponential factor that you do not mention, and there is no simple formula for the nontrivial zeros, so it is dubious that you can estimate the contribution from all the nontrivial zeros using tables of approximations of some zeros. Why are you insistent on computing the zeta-function by this approach? – KCd Mar 07 '18 at 14:51
  • I have a signal-processing background and therefore I tend to view the complex values of zeta on the critical line as the complex frequency response of some unknown system. Therefore I am trying to get some intuition by observing the contribution of the poles and zeros to the values on the critical line. For values of w that are small (less than the first few hundred zeros), there are enough published zero locations that I can compute with sufficient accuracy for my purposes. – Bob Mar 07 '18 at 14:56
  • The Hadamard product uses exponential factors to allow convergence. Look up the Hadamard product for the Gamma function. In fact, you can essentially combine the contribution from the trivial zeros for $\zeta(s)$ into a single factor $\Gamma(s/2)$ and work on computing that along the critical line. – KCd Mar 07 '18 at 15:16
  • Thanks, that should work. – Bob Mar 07 '18 at 15:19

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