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According to this website, we see that Euler formally proved (observed?) his result connecting the harmonic series and a product over primes. Riemann wrote his historical paper in 1859 which considers the same observation, but now with $$\zeta(s)=\sum_{n=1}^\infty n^{-s}=\prod_p\left(1-p^{-s}\right)^{-1}$$where $s\in\Bbb C$. According to the same website above, it is said that Kronecker proved in 1876 the identity $\sum_{n=1}^\infty n^{-s}=\prod_p\left(1-p^{-s}\right)^{-1}$ for $s>1$. If this is the case, how well was Riemann's paper received by modern mathematicians at the time? It would seem to be based on something unproven. Or is it more likely something that all of the professional mathematicians would find obvious; hence, not worth publishing?

I've attempted to find some references regarding Kronecker, but MathSciNet didn't pull anything up (the publications jump from 1869 to 1881).

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Here is Wilkins's translation of Riemann's paper. Remarks at the start of the paper show that Riemann was well aware of the convergence issues:"For this investigation my point of departure is provided by the observation of Euler that the product $\prod\frac{1}{1-\frac{1}{p^{s}}}=\sum\frac{1}{n^{s}}$ if one substitutes for $p$ all prime numbers, and for $n$ all whole numbers. The function of the complex variable $s$ which is represented by these two expressions, wherever they converge, I denote by $\zeta(s)$. Both expressions converge only when the real part of $s$ is greater than $1$; at the same time an expression for the function can easily be found which always remains valid". The reference here is not to Euler's 1737 "factorization" of the harmonic series but to 1748 Introductio in Analysin Infinitorum, where the identity for $s>1$ appears. Although Euler did not work with convergence in the modern sense he already knew the difference between $s=1$ and $s>1$ cases since he summed the Basel series in 1735 ($s=2$, later all even $s$), and used unlimited growth of harmonic series to prove infinitude of primes in that 1737 paper.

Cauchy addressed convergence of the $p$-series for real $s$ in his texts, and the infinite product reduces to a sum by logarithming, a trick used by Riemann later in the paper, and by others before him. In fact, Dirichlet used analogous identity for $L$ functions, with a character $\chi$ on top of the fractions, in his famous 1837 paper on primes in arithmetic progressions. And in 1848 Chebychev presented to St. Petersburg Academy his results, published in early 1850s, on the asymptotics of the count of primes (a version of the "prime number theorem"), where he logarithms the Euler product for real $s>1$, and derives an identity for $\zeta(s)$ (not under this name, of course). In 1859, when the paper was published, Riemann was familiar with the work of Dirichlet and likely Chebychev (who is mentioned in his unpublished notes), so he saw replacing real $s$ with complex one, and rephrasing the convergence in terms of real parts, as a minor issue. It appears that Kronecker simply wrote down what was a well known folklore.

The main insight of the paper is a close relation discovered by Riemann between specific traits of the distribution of primes and locations of the zeros of the $\zeta$ function, which lead to strengthened formulations of the prime number theorem. The problem was of high interest to contemporary mathematicians, with Gauss and Bertrand in addition to Dirichlet and Chebychev making important contributions prior to Riemann's work. The paper redirected the further work on the prime number theorem into analytic vein leading to Hadamard's and Vallee Poussin's 1896 results on $\zeta$ zeros. See Bombieri's exposition in the millenium problem description.

This being said, the standards of rigor weren't modern ones even after Gauss. In another paper Riemann used the Dirichlet principle to prove existence of a holomorphic function with specified poles on a Riemann surface, the kind of variational existence argument later famously criticized by Weierstrass. According to Klein's Development of Mathematics in the 19th Century, this did not prompt anyone, Weierstrass included, to doubt his results. To paraphrase, rigor is a friend but insight is a better friend. This hasn't changed if Witten's Fields medal and the vast literature on mirror symmetry are any indications. Perelman left out many technical steps and conspicuously declined to publish his proof of geometrization in refereed journals, and so on.

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    $\begingroup$ This is very well-written; I appreciate it greatly. I hadn't been able to find a reference for the identity appearing earlier than Kronecker's publication. +1; thanks so much! $\endgroup$ – Clayton Sep 12 '15 at 20:24
  • $\begingroup$ Thanks for the updated title; it is more fitting, I believe. $\endgroup$ – Clayton Sep 15 '15 at 18:14
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The formula you wrote is indeed due to Euler. Whether he did have a rigorous proof or not is not important. At the time of Riemann giving a rigorous proof of this was a relatively easy exercise.

Riemann's famous paper contains MUCH MORE than this formula, which is only a starting point. You should look to his paper (or to the enormous literature generated by it) before asking such question. Open any book titled Analytic number theory. It will contain an exposition of this paper, many of these books in English.

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  • $\begingroup$ Fortunately, I'm doing my Ph.D. in analytic number theory, so I am quite well-acquainted with the field, which is why I found this question interesting. // I'm not asking if Euler gave a rigorous proof; I'm asking if a rigorous proof had been given by the time of Riemann or not. (1) If not, why was Riemann's paper met with such wide acclaim? If I were to write a paper based on an unproven statement, I would likely be asked by the referee to provide a proof of the statement. (2) If it had a rigorous proof, why did Kronecker publish the proof of the theorem nearly 20 years later? $\endgroup$ – Clayton Sep 11 '15 at 21:22
  • $\begingroup$ For the interested reader, a link to the English-translation version of the paper (due to David R. Wilkins) can be found here. $\endgroup$ – Clayton Sep 11 '15 at 21:42
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    $\begingroup$ Riemann's paper had a wide acclaim because of his new results, not because of this formula. Namely the functional equation for zeta function, and for the whole idea to use complex analysis. Which led to the proof of the asymptotics by Hadamard and Valle-Poussin, $\endgroup$ – Alexandre Eremenko Sep 12 '15 at 1:42
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    $\begingroup$ If I were a referee, I would recommend to publish anything Riemann wrote, no matter with proofs or without:-) $\endgroup$ – Alexandre Eremenko Sep 12 '15 at 17:59
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    $\begingroup$ @Alexendre Emerenko:Lol I like such humour. $\endgroup$ – Nicco Sep 12 '15 at 19:40

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