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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
    Commented Mar 20, 2020 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? $\endgroup$ Commented Mar 21, 2020 at 22:04

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Riemann wrote his paper on the zeta-function in 1859. He was the first person to consider the zeta-function in the complex plane, first for ${\rm Re}(s) > 1$ and then he worked out an analytic continuation to all of $\mathbf C$ except for a simple pole at $s = 1$. Before Riemann the only way the zeta-function was used was on the real line: real numbers bigger than 1 for Chebyshev and Dirichlet, and positive and negative real numbers by Euler using divergent series. Riemann's use of complex analysis in his paper on the zeta-function was unlike anything anyone had done before. For example, his use of Fourier inversion to replace an integral on the positive real line with a vertical integral in $\mathbf C$ was new.

There were three majors figures in the development of basic complex analysis: Cauchy (1810s-1850s), Riemann (1850s-1860s), and Weierstrass (1850s-1880s). All Riemann had available to learn complex analysis was the papers of Cauchy, which were rather disorganized since Cauchy himself did not realize until the 1850s the many ways that complex-valued functions of one complex variable were far more special than functions of two real variables. The main ideas Cauchy had were his integral formula and his residue theorem, but in a much more primitive form than the way we know them today.

Riemann figured out a lot on his own: his dissertation in 1851 was about Riemann surfaces, going far beyond what Cauchy had been doing. Cauchy's focus was on using contour integrals and residues in the complex plane, mainly to compute real integrals. Riemann's vision was totally different and in many ways much grander: use the Cauchy-Riemann equation (PDEs) and conformal mappings, analytic continuation, and study functions geometrically rather than by relying on formulas for their values everywhere. Weierstrass did everything with power series rather than with integrals or geometry. The ground in complex analysis that was prepared to let Riemann conjecture the Riemann hypothesis had been developed in large part by Riemann himself.

See Section 5.10 of Bottazzini and Gray's "Hidden Harmonies - Geometric Fantasies: The Rise of Complex Function Theory" for a summary of Riemann's paper and a description of what was novel in it.

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As a further data point, you might want to look at André Weil's article "On Eisenstein's Copy of Disquisitiones" (Advanced Studies in Pure Mathematics 17, 1989, pp. 463-469), where he suggests that Eisenstein might have influenced Riemann when he wrote his paper on the zeta-function.

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    $\begingroup$ Weil wrote in that paper that Eisenstein had an unpublished proof of the functional equation of $L(s,\chi_4)$ using Poisson summation, which Weil thinks may have inspired Riemann in using Poisson summation. In contrast to the initial definition of $\zeta(s)$, note that $L(s,\chi_4)$ and $L(1-s,\chi_4)$ both make sense for real $s$ in $(0,1)$ from the original definition of $L(s,\chi_4)$ as a series, so you can consider a functional equation relating values at $s$ and $1-s$ from the definition as a series. That is not the case for $\zeta(s)$ as $\sum 1/n^s$. $\endgroup$
    – KCd
    Commented Aug 17, 2022 at 16:02

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