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For the purpose of formalisation, I am looking for a simple proof of the function equation of the Riemann ζ function or the generalisation thereof, the Hurwitz's formula for the Hurwitz ζ function.

There are two approaches that I am aware of:

  • Using a contour integral and Cauchy's Residue Theorem
  • Using Poisson summation (e.g. via Lipschitz summation or the θ function)

The first one will probably be quite messy because the contour that is used has one segment from -∞ to -ε and one from -ε to -∞ (i.e. the same path, but in opposite direction) and on one of them, one branch of the logarithm is taken and on the other, another branch. It seems to me that to formally justify this, one has to shift the paths away from the real axis by some small δ and consider the limit, which seems a bit messy.

The more popular approach these days seems to be through Poisson summation, but that probably requires Fourier series and Fourier transforms and possibly results on Lp spaces as well.

Is there an easier approach to tackle this? Or is there a nice and easy way to make the contour integration approach rigorous?

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  • $\begingroup$ Gamelin's complex analysis textbook describes the contour-integral approach, and that particular presentation is not terribly difficult or messy IMO. $\endgroup$
    – james h
    Commented Jul 29, 2017 at 20:55
  • $\begingroup$ Unfortunately, that also uses this idea of integrating along a branch cut and taking a different branch each time. I am not saying it's wrong to do that, but I do think it should be justified explicitly, and I cannot find such an explicit justification anywhere. $\endgroup$
    – Manuel Eberl
    Commented Jul 30, 2017 at 8:59
  • $\begingroup$ You should edit the question to clarify what you're really interested in: Riemann or Hurwitz? The book by Titchmarsh (Theory of the Riemann Zeta-function) offers seven different proofs; none requiring Lipschitz summation or Lp spaces. On the other hand, Hurwitz zeta is less central to the subject and more difficult. $\endgroup$
    – Stopple
    Commented Jul 31, 2017 at 14:33
  • $\begingroup$ Well, both are of interest to me. If possible, Hurwitz would be more interesting, since it is more general (the equations for Riemann ζ and Dirichlet L-functions easily follow from it). In any case, I now think the ‘standard’ contour integral approach might be feasible for me after all, so I'll see if that works out. $\endgroup$
    – Manuel Eberl
    Commented Jul 31, 2017 at 20:12

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