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In the Chapter 7 epigraph in Stein and Shakarchi's Complex Analysis, there is a quote, apparently from Hadamard in 1945, saying

He proved some important properties of that function, but enunciated two or three as important ones without giving the proof. At the death of Riemann, a note was found among his papers, saying “These properties of ζ(s) (the function in question) are deduced from an expression of it which, however, I did not succeed in simplifying enough to publish it.” We still have not the slightest idea of what the expression could be. As to the properties he simply enunciated, some thirty years elapsed before I was able to prove all of them but one. The question concerning that last one remains unsolved as yet, though, by an immense labor pursued throughout this last half century, some highly interesting discoveries in that direction have been achieved. It seems more and more probable, but still not at all certain, that the “Riemann hypothesis” is true.

I tried searching online, but haven't found anything about this mysterious expression, or what the two or three properties are that Riemann left unproven and that Hadamard did successfully prove. Are there any sources/details for this quote?

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  • $\begingroup$ For zeta function, maybe: J. Hadamard, Etude sur les Propriétés des Fonctions Entières et en Particulier d’une Fonction Considérée par Riemann (1893) or Sur la distribution des zéros de la fonction ζ(s) et ses conséquences arithmétiques (1896) $\endgroup$ Feb 8, 2022 at 11:22
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    $\begingroup$ The quote is from Jacques Hadamard, The Psychology of Invention in the Mathematical Field (1945), English transl. page 118 $\endgroup$ Feb 8, 2022 at 11:22

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The mysterious expression, that Riemann did not "simplify enough to publish", is mentioned in his letter to Weierstrass published by Weber in 1876. It is reminiscent of Fermat's "marvelous proof" that the margin was "too narrow to contain", but a review of Riemann’s Nachlass led a librarian (Distel) to the likely culprit, apparently, unbeknownst to Hadamard. Siegel gave a reconstruction of it in On Riemann’s Nachlass for Analytic Number Theory (1932), and it is now known as the Riemann–Siegel formula:

"In fact, Herr Distel, a librarian, already several decades ago discovered the representation in question of the zeta function in Riemann’s papers. It concerns an asymptotic development which gives the behavior of the function $ζ(s)$ on the critical line $σ = 1/2$ and, more generally, in each strip $σ_1 ≤ σ ≤ σ_2$, expressed for infinitely increasing magnitude of $s$. The principal term of this development was meanwhile rediscovered, independently of Riemann, by Hardy and Littlewood in 1920 as a consequence of their “approximate functional equation”; they use the same methods of proof as Riemann, namely approximate calculation of an integral by the saddle-point method. In Riemann, however, there is also a process for obtaining additional terms of the asymptotic series, which is based on the beautiful properties of the integral $$Φ(τ, u)=\int\frac{e^{\pi i\tau x^2+2\pi iux}}{e^{2\pi ix}-1}dx,$$ which, incidentally, has also led Kronecker and recently Mordell to a most elegant derivation of the reciprocity formula for the Gauss sums.

In 1926 Bessel-Hagen noted in a new review of the Riemann papers another previously unknown representation of the zeta function in terms of definite integrals; in this Riemann was also guided by the properties of $Φ(τ, u)$. These two representations of $ζ(s)$ may be included among the most important results in Riemann’s number-theoretic Nachlass, insofar as one does not find them in his published paper."

The first "two or three" properties that Riemann derived from these expressions are likely that the Riemann xi function $\xi(s)$ is an entire function of $s^2$, with a Weierstrassian product expansion, which Hadamard rigorously proved in 1893, and an asymptotic form of the prime number theorem that he derived from it in 1896, see Iurato, On some historical aspects of the theory of Riemann zeta function, pp. 77-78, 104. Both are traceable to Riemann's 1859 paper, and Iurato specifically alludes to Hadamard's remark from The Psychology of Invention in the Mathematical Field (1945) in connection with them:

"Following (Stopple 2003, Chapter 10, Section 10.1), it was Riemann to realize that a product formula for ξ(s) would have had a great significance for the study of prime numbers. The first rigorous proof of this product formula was due to Hadamard but, as he himself remembered, it took almost three decades before he reached to a satisfactory proof of it. Likewise, also H.M. Edwards (1974, Chapter 1, Sections 1.8-1.19) affirms that the parts concerned with (2) are the most difficult portion of the 1859 Riemann's paper (see also (Bottazzini & Gray 2013, Chapter 5, Section 5.10)). Their goal is to prove essentially that $ξ(s)$ can be expressed as an infinite product... "Hadamard (in 1893) proved necessary and sufficient conditions for the validity of the product formula $ξ(t) = ξ(0) ∏_ρ(1 − t/ρ)$ but the steps of the argument by which Riemann went from the one to the other are obscure, to say the very least"...

Just in regard to the novelties due to Hadamard, Weyl points out that the driving force for these Hadamard's investigations was the wish to obtain sufficient information about the zeros of the Riemann zeta function for establishing the asymptotic law for the distribution of prime numbers... In 1896, both Hadamard and de la Vallée-Poussin, independently of each other, were able to draw the conclusion $\left[\frac{\pi(n)\ln(n)}{n}\to1\right]$ from 1893 Hadamard's results concerning entire functions".

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    $\begingroup$ Absolutely brilliant detective work! Do you mind sharing a bit behind your process, like what major queries you searched online, or the general approach you took to find all this out? $\endgroup$
    – D.R
    Feb 9, 2022 at 8:11
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    $\begingroup$ @D.R First I looked at comprehensive works on the history of Riemann zeta function, like Iurato's and Edwards', with an eye for Hadamard's contributions. His quoted timeline helped identify the matching ones, and Iurato's remark confirmed it. Then I looked for the source of Riemann's quote and references to it, which brought up the Siegel's paper. $\endgroup$
    – Conifold
    Feb 9, 2022 at 8:20
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    $\begingroup$ Davenport's Multiplicative Number Theory lists in Chap. 8 five "conjectures" from Riemann's paper (I am not claiming Riemann himself thought all were unproved, but later mathematicians definitely saw some gaps to fill in): (a) infinitely many nontrivial zeros, (b) an estimate for $N(T)$, which is the number of nontrivial zeros with multiplicity having imaginary part from $0$ to $T$, (c) a product over the zeros for the entire function $\xi(s) = (1/2)s(s-1)\pi^{-s/2}\Gamma(s/2)\zeta(s)$, (d) an "explicit formula" for something like $\pi(x)$ in terms of nontrivial zeros, (e) RH. $\endgroup$
    – KCd
    Feb 10, 2022 at 3:32
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    $\begingroup$ The estimate for $N(T)$ has main term $(T/2\pi)\log(T/2\pi)$, so (b) implies (a). Riemann perhaps believed he had a proof of (b), and thus also (a). Maybe he believed he had a proof of (c). Definitely he did not have a proof of (d), which you could consider to be a form of the prime number theorem. So (d) and (e) are at least two results Riemann did not claim to have proved. The purpose of Riemann's paper was to sketch a program that could lead to a proof of the prime number theorem. $\endgroup$
    – KCd
    Feb 10, 2022 at 3:37
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    $\begingroup$ What Riemann proved in the paper and nobody after him disputed (but people did gradually find alternate proofs for them anyway) was the analytic continuation and functional equation for $\zeta(s)$. $\endgroup$
    – KCd
    Feb 10, 2022 at 3:40

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