Yuri Manin, in a translation of a recent interview, available in the forthcoming issue of the Notices of the AMS:
I’m somewhat apprehensive that its [the Riemann Hypothesis] first solution might be a proof using blunt analytic methods. It will receive every imaginable prize, the solution will be acclaimed in every newspaper in the world, and all of this will be misleading because the “right” solution should be given in a wider context, which we already know. We even know several approaches to a solution.
(Note: I do not speak or read Russian, so I can not check the accuracy of the translation from the original article).
Yes, the translation is accurate. By the way, the interviewer is a grandson of late Israel Gelfand.
If anything, the statement has been toned down in the English translation. In Russian, he did not say “somewhat apprehensive” but “very fearful” (я очень боюсь), not “misleading” but “stupid/asinine/silly” (глупость), not “should” but “must only” (должен только), and the word he used for “blunt” (тупой) carries connotations of dullness/stupidity/obtuseness as well.
я очень боюсь I am very afraid
глупость stupidity
тупой dull/stupid
I understand the concern, but I can’t really see any situation in which having a proof of RH is worse than not having one.
If the “blunt analytic method” extends to prove GRH and ERH, and perhaps also an independent proof of the function fields RH, then it is close enough to the “right” proof for me, much as the complex-analytic proof of the PNT is. If it instead relies on some transient property of the integers that is not shared by other number fields etc., then that would also be very interesting, and would no doubt inspire a huge amount of effort to work on these other cases. Even if the analytic approach does not directly work for, say, GRH, it will presumably provide some level of insight and heuristic guidance.
In any event, given how easy it is to perturb the zeta function to create a zero off the critical line while staying consistent with all the known analytic facts about zeta, it seems unlikely that RH will be proved without using intimately using the properties of the integers (and not just unique factorisation (Euler product), uniform distribution (pole at s=1), and Poisson summation (functional equation)).
Cet extrait de l’article d’Emile Borel “Sur l’intégration des fonctions non bornées et sur les définitions constructives” exprime parfaitement mon sentiment sur ce genre d’opinion. (I apologize for Lagrange’s political uncorrectness).
“Mon maître Jules Tannery citait volontiers une phrase de Liouville; après avoir comparé les démonstrations longues aux démonstratioîis
courtes, il concluait : “En somme, les démonstrations longues ont un grand avantage, c’est d’être longues, et les démonstrations courtes ont un grand avantage, c’est d’être courtes”. Et Liouville rappelait, paraît-il, quand il était de bonne humeur, la boutade qu’on attribue à Lagrange: « Les mathématiques sont comme le cochon, tout en est bon”.
Here is an approximate English translation of Borel’s quote:
“My teacher Jules Tannery liked to quote Liouville; after comparing short proofs and long proofs, he used to conclude: ‘So, long proofs have one big advantage: they are long; and short proofs have one big advantage: they are short’. And, people say, where he was in a good mood, Liouville recalled the joke that is attributed to Lagrange: <>”
I find the previous sentence in the same article even more objectionable.
“The Riemann Hypothesis, without a doubt, is a problem that Riemann originated within a program, although during the course of a century and a half, the narrow number theorists continued to look at it as a very important isolated challenge.”
Narrow or not, I don’t think any number theorist would associate the word ‘isolated’ with the Riemann Hypothesis.
I heard similar statements about the proof of the Poincare conjecture. So now people are looking for the “right proof”.
yes it’s true that someone in russia has provided grounds to a proof RH but due to export restriction on this and other material it will never be published nor available
I appreciate Terence Tao’s comment. Perhaps he or someone could elaborate on why RH would likely need more arithmetic information? For example, I’d to know what analytic properties can be preserved by perturbing the zeta function.
And, yes, when I read the Notices article I didn’t think too much about the above quote as I knew what Manin meant. But now that you discuss it here, I think he (Manin) might have been a little overboard about it. I for once would be thrilled to know there’s a proof no matter what type of proof it is. RH is so beautiful with so many important consequences that just knowing it’s true would be a great feeling.