I’m exploring the history behind the Riemann Hypothesis and I found something interesting. We know that Bernhard Riemann was mainly focused on complex analysis, but he also wrote a very important paper on number theory, where he introduced the Riemann Hypothesis. It seems this was his only work in number theory.

I’m curious about why he decided to write this paper. I read on some websites that he was asked to prepare a report for the Prussian Academy of Sciences. It’s said that his former teacher, Carl Friedrich Gauss, gave him this task. Is this true? If yes, where can I find more information about it? I’m really interested in learning more about why he shifted his research focus.

Thanks for any help you can provide!

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    $\begingroup$ If I remember correctly, Gauss asked Riemann to write a treatise on one of three topics (which Riemann had to choose from): foundations of geometry, convergence of trigonometric series, and a third topic which I don't remember exactly but I am pretty sure it was not related to number theory. Riemann's work in number theory was not a fruit of a task suggested by Gauss. There are speculations that Riemann's work was influenced by Eisenstein; please read hsm.stackexchange.com/questions/11565/… . $\endgroup$
    – user2554
    May 12 at 11:52
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    $\begingroup$ If you read this paper of Riemann, you see that it is also essentially in complex analysis. And also the first major advance after Riemann (the proof of the asymptotic law by Hadamard and Vallee Poussin) was achieved by analysts. This part of number theory, called Analytic number theory is essentially a chapter of Complex Analysis. $\endgroup$ May 12 at 11:56
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    $\begingroup$ The "task" given by Gauss concerned Riemann's 1854 lecture on the foundations of geometry, not the Riemann Hypothesis paper. Gauss died in 1855, the paper came out in 1859 after Riemann's visit to Paris where he met Bertrand and Hermite, among others, also interested in number theory. In fact, Riemann's interest in it goes back to his 1849 Berlin stay where he studied it with Dirichlet and Eisenstein. Generally, Riemann's interests extended far beyond complex analysis, see Tazzioli's short survey. $\endgroup$
    – Conifold
    May 12 at 21:43


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