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Although I do not doubt in Riemann˙s originality, I would like to know how much complex analysis was developed up to the day when Riemann conjectured what is today called Riemann hypothesis and how much itself was the theory of zeta-functions developed up to that day?

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    $\begingroup$ This is a bit broad for this site, but I think all the information you want is contained in the History section II of the Bombieri's description of the Millenium problem. If not, perhaps you can narrow down the question after reading it. $\endgroup$ – Conifold Mar 20 at 23:27
  • $\begingroup$ Apart from other aspects, it is my impression that "complex numbers" and/or "complex analysis" were partly viewed as "convenient fictions" in those years, so, to my mind, a great part of the conceptual novelty was that non-real zeros of functions had very tangible (non-dismissible) real-life impacts. Such broader-context-issues were not addressed so much in Bombieri's essay. Is the complex-analysis context what you're wanting, specifically? $\endgroup$ – paul garrett Mar 21 at 22:04

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