When Bernhard Riemann died, his housekeeper trashed several papers written by him. Probably among that documentation which could not reach to belong Riemann's Nachlass, was 'a small black book containing records of Riemann's sojourn in Paris in spring 1860' (page 153, "The music of the Primes", Marcus Du Sautoy, first edition, 2003). Now, whatever Riemann might have written during his stay in Paris, certainly was after he published his 1859 paper "über die anzahl der primzahlen unter einer gegebenen grösse", which contains what nowadays is known as "The Riemann hypothesis" (RH). It would seem suggesting that Riemann perhaps continued some of the ideas arisen in the aforementioned paper. Assuming this last can be concluded, we are now 155 years after the publication of the Riemann hypothesis. Much has been worked since then. I am not saying is the same case as "Fermat's last theorem", which was actually proven 358 years later by Andrew Wiles, supported by mathematical ideas far away of what Pierre de Fermat might have possibly be acquainted with. Having said that, it remains as a fact that there has been much work since Riemann's 1859-paper, so, the question is, What is the real importance of the missing black book? if it were here with us now, would it make a difference in the research of the RH? if so, in what particular direction? or (thinking romantically for a moment) would it show to be critical for solving RH?
THis black box might be as interesting as Ramanujan lost notebook; but probably useless to solve RH. If Riemann himself had a proof of RH, mathematicians of his generation or a little later would have found it. Just like if Fermat had a proof of FLT etc..
I share the opinion of some mathematicians who accepts that the RH
can not be solved by traditional methods of analysis. The proof must based
on a new idea.
Article The Riemann Hypothesis
Riemann did not explain his ideas for solving problems. C.L.Siegel using the analytic number theory to explain the geometric problem, can not find the idea of geometry.
We use the computer to draw the geometry of the zeta function and obtain its the the geometries law,on this basis, we have completely restored his manuscript. Namely, Riemann himself has been in a very clever way to prove the "Riemann hypothesis."
“If you can’t explain simply, you don’t understand it well enough.” — Albert Einstein.
What does the Riemann Hypothesis (RH) mean?
RH confirms the existence of prime numbers in an optimal way. Or rather, for all positive integers, k > 1, there exists a prime number, p, which divides k such that either p = k or p ≤ sqrt(k) = k^(1/2) where RH states the exponent of k is 1/2.
Please keep that fundamental fact in mind when discussing the truth of RH.
Reference link:
https://www.researchgate.net/post/Why_is_the_Riemann_Hypothesis_true2;
https://www.researchgate.net/post/Does_the_nth_nontrivial_simple_zero_of_the_Riemann_zeta_function_indicate_the_nth_prime_p_n_occurs_as_a_prime_factor_in_all_multiples_of_p_n.
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