Talk:Riemann hypothesis/Archive 4

Landau function

Recently an unsourced claim was removed that I seem to remember finding (independently) in some reliable source. Roughly it just says that the RH hypothesis is equivalent to a sharp estimate on yet another prime counting function, so it is not particularly exciting. However, I could not find the claim explicitly in any sources this time when I looked. The same claim is at Landau function, and so should probably be sourced at some point. The difficulty is that Landau did work on stuff related to the Riemann hypothesis, and many articles on Landau's function do not mention Landau, so I have had a difficult time phrasing the search. Landau's function is also called "the maximal order of a permutation" or "the maximal order of an element of the symmetric group" and can be closely approximated by a simply weighted prime counting function. Sometimes this part is called "statistical group theory" or a part of "asymptotic group theory".

At any rate, I would be interested if someone managed to track down/remember this reference. It is not hard to find the implication "RH ⇒ Bound", but the other direction was not relevant in any of the papers I found. JackSchmidt (talk) 17:28, 7 February 2009 (UTC)

The fact was added a long time ago. I asked the original contributor for his source, but I think someone else asked back in March and the editor seems somewhat retired, though editing every few months. JackSchmidt (talk) 00:59, 8 February 2009 (UTC)

150ths birthday and FA?

At WT:WPM, I recently suggested improving this article to FA class. We could then display it at the main page in November (Riemann published the paper in Nov. 1859). Is anybody interested in doing that joint effort? Jakob.scholbach (talk) 11:17, 8 February 2009 (UTC)

Farey sequence

Resolved

The equivalence of RH with an estimate on the Farey sequence was recently removed as unsourced. However, this section is trivial to source, for instance MR2199580 has this as its title. MR2105342 states that, "It is well known that the famous Riemann hypothesis is related to estimates of error terms associated with the Farey sequence of reduced fractions in the unit interval." There are several other papers in this vein found on the very attempt (indeed, I noticed this because I found several of them trying to source the RH ⇔ Landau function estimate). One of the earliest I have found is MR0028354, but I have not checked any sort of historical sources, this is just one of the many hits on mathscinet.

I do not object to the removal since I think the article is being cleaned up nicely, only to the edit summary. Unsourced material should not be removed (for that reason) unless the material is disputed, and I think the disputer should make a good faith effort to verify the fact. For instance, I'm feeling lucky on google scholar for Farey Riemann works well.

At any rate, assuming the material was removed to help with some of the WP:UNDUE weight being placed on equivalents (that are more likely corollaries than proof methods), I support the removal, but if it really was removed for being unsourced, then one could add 10 or 20 sources easily to support the claims. JackSchmidt (talk) 14:39, 10 February 2009 (UTC)

Looking a little at the history, I think this goes back to Helge von Koch around the turn of the century. It might therefore be worth including at some point, given the number of papers developing the idea over more than 100 years. JackSchmidt (talk) 16:50, 10 February 2009 (UTC)

The material has now been included nicely in the new article. It does not mention the amount of work done on this problem, but I think that is a good thing. The more detailed development of this research line can be done in a separate article on the Farey sequence or even in an article on the relation between the Farey sequence and the Riemann hypothesis if there is more material than I found. The only part that might be interesting enough to adjust this article is if Koch had done work on this earlier than 1924. JackSchmidt (talk) 17:49, 14 February 2009 (UTC)

Gram blocks

I added a {{clarify me}} tag to the paragraph about Gram blocks, which R.e.b. removed. I have the following specific questions: (1) A Gram block is an interval, right? The paragraph defines it in a way that looks like a set of two individual points, but I think this is just incorrect. What I'm less clear on is whether the block consists of only the points on the critical line between the Gram points, or whether it consists of a rectangle of the whole critical strip. (2) What is "the expected number" of zeros in one of these intervals? Two (assuming that there's one bad point between every two good points), or is it the number of bad points plus one? (3) What is the technical meaning of "most blocks" here? I would assume that it means that as x->infinity the proportion goes to one, but I can imagine other possible meanings. (4) What does it mean to say that some blocks "may not have a unique point"? My best guess is that it means exactly one bad Gram point, but it's not obvious. —David Eppstein (talk) 18:01, 12 March 2009 (UTC)

Thanks, now it's much clearer. —David Eppstein (talk) 19:53, 12 March 2009 (UTC)

Equivalence

I suspect this is well known, but I had trouble verifying the given sources: Does Koch or Schoenfield actually prove that the RH is equivalent to the "best possible" bound for the error of the prime number theorem (or the more precise result). I think I can say with some certainty that Schoenfield 1976 does not, and I could not find such a claim in Koch 1901. I believe both only state consequences of the RH, not sufficient conditions for the RH to hold. However, neither my French nor my analytic number theory is equal to the task; perhaps the other direction is trivial. Is there a clear statement of the equivalence in a reliable source, even a secondary source? JackSchmidt (talk) 17:52, 13 March 2009 (UTC)

Hmm. It seems to be so well known that nobody cares to give a useful reference: it is mentioned e.g. in [1], [2], [3], without source. Wolfram gives one reference (Wagon 1991) in [4], and a whole bunch of them near the end of [5]. I suspect that there may be something useful in "The Theory of the Riemann Zeta-function" by Titchmarsh & Heath-Brown. Now we need someone with a good library and time to spare to check it. — Emil J. 18:26, 13 March 2009 (UTC)

Cool. Personally, I am satisfied with these (unsourced) corroborating sources (claymath one seems fine, for instance). The current section reads better without referring to any of these, but it is nice to know it was easy to find such a statement in probably the most obvious places to look. If someone (with a good library and time to spare) can check for a mathematically and wikipedia-ly satisfying secondary source, then it might be worth noting here at the very least, but for now the current sourcing seems just fine. JackSchmidt (talk) 19:16, 13 March 2009 (UTC)

Robin's theorem consensus

In addition to the previous question of equivalence (resolved as "yes, they are equivalent"), there seems to be another repeated edit involving adding a source and a restriction to even (or odd, depending on the edit) n to Robin's theorem.

The restriction part of the edit seems clearly wrong to me, as it immediately fails verification from the source. The source itself seems not to meet the standards currently in this article, since it is just a preprint. Rather we should cite Robin, which we already do. It seems to me that both sources (the preprint and Robin) confirm the statement for all n, and that even if the claim for even or odd n happens to be true, it is better to stick with Robin's language.

Do we agree that the new source is not appropriate and does not support the restriction to even or odd n in Robin's theorem, and that in any event we already cite the authoritative source (Robin) and that this source verifies the claim without restriction on n? JackSchmidt (talk) 17:20, 16 March 2009 (UTC)

The person who made the above edit (making the restriction to even n) was User:Katsushi. It has been suggested over at WT:WPM#WAREL/DYLAN LENNON that he is a sock of User:WAREL. In his entire career on Wikipedia, Katsushi has never left a Talk comment. EdJohnston (talk) 17:47, 16 March 2009 (UTC)

The new source citation was improved, and the math review's abstract made the edit more clear. Robin proved that RH was equivalent to the inequality being true for all n ≥ 5041, and MR2394891 proves unconditionally that the inequality holds for many n, including all odd n. Thus RH iff inequality for all n ≥ 5041 iff inequality for all even n > 5041 iff inequality for all even n divisible by the 5th power of a prime.

Perhaps one could have:

---

The Riemann hypothesis is equivalent to many other conjectures about the rate of growth of other arithmetic function aside from μ(n). A typical example is Robin's theorem (Robin 1984), which states that if σ(n) is the divisor function, given by

$\sigma (n)=\sum _{d\mid n}d\,$

then

$\sigma (n)<e^{\gamma }n\log \log n\,$

for all n > 5040 if and only if the Riemann hypothesis is true. It has been shown that the inequality holds for large odd and square-free integers, and that the Riemann hypothesis is equivalent to the inequality just for n divisible by the fifth power of a prime, (Choie et al. 2007).

• Choie, YoungJu; Lichiardopol, Nicolas; Moree, Pieter; Solé, Patrick (2007), "On Robin's criterion for the Riemann hypothesis", Journal de théorie des nombres de Bordeaux, 19 (2): 357–372, arXiv:math/0604314, ISSN 1246-7405, MR 2394891 {{citation}}: Cite has empty unknown parameter: |1= (help)

---

It would be nice to shrink the addition a little, and to check for additional important papers, but this sort of thing looks fine to me. What do other's think? JackSchmidt (talk) 19:33, 16 March 2009 (UTC)

The question to me is not so much whether the new statement is accurate (I think it is), but whether it is significant among all the other variations of statements like this that one could write of problems equivalent to Riemann. Is proving restrictions of this form on the potential counterexamples likely to make progress towards Riemann itself? If so, I'd say it's worth including. But if it's not directed towards an eventual proof of Riemann, statements like this that say that a counterexample must have certain specific properties seem to me much like counting angels on heads of pins: what good is it if one doesn't believe the angels even exist? I tend to feel that we should stick to a single clean statement (Robin's original theorem) rather than cluttering it up with cruft. But I note that R.e.b. has removed other more significant variations in some of his past edits (e.g. the Lagarias 2002 one that Riemann is equivalent to $\sigma (n)\leq H_{n}+\ln(H_{n})e^{H_{n}}$) and I would be happy to defer to his judgement on this one as well. —David Eppstein (talk) 20:42, 16 March 2009 (UTC)

Sounds good to me too. Currently Robin's theorem is a redirect to a page that only briefly discusses it. Perhaps that page can have a dedicated section on Robin's theorem (which will improve the redirect), and this sort of addition would fit better there. Does that sound reasonable? The material is worth including in the encyclopedia, but under a more relevant article? JackSchmidt (talk) 21:11, 16 March 2009 (UTC)

Actually, I'm having trouble deciding where to put the section breaks on divisor function. I think "Approximate growth rate of the sigma function" is the right level of granularity for the section (not "Robin's theorem"), but actually choosing the section heading is hard, and needed before the redirect can be fixed. I included the refs for Lagarias 2002 over there too. Part of making this RH article great is making the articles it links too at least good. There is no sense including a link to an article which only has the same sentence repeated somewhere near the bottom. JackSchmidt (talk) 21:37, 16 March 2009 (UTC)

Making Robin's theorem into a real article rather than a redirect, with more detail about this sort of variation, and keeping it out of the main Riemann article, sounds like a good solution to me. I agree that we want to have the surrounding articles and not just the main Riemann article be in decent shape. I note that in the current article that Robin redirects to, it states that counterexamples must be superabundant, but that the comment on the abstract page for arXiv:math/0604314 seems to be retracting this claim. Is there a different source for the superabundance of these numbers? —David Eppstein (talk) 22:15, 16 March 2009 (UTC)

Good catch. I tried to give citations to each of the other claims and removed the superabundant one. The published Choie et al. paper had a reasonable history of the problem (to my untrained eye), and I included the 1915 Ramanujan result. After doing all this reference work I remembered that I personally only wanted estimates on "d", σ₀, so oops, read a bunch of irrelevant number theory. JackSchmidt (talk) 23:29, 16 March 2009 (UTC)

The proof by Choie et al. (in the first version of the article on arXiv), that the smallest counterexample to Robin's inequality must be superabundant, seems to me to be correct. (The error was to assume that the smallest 5-free counterexample also must be superabundant.) Another (very similar) proof of this result is included in the note Superabundant Numbers and the Riemann Hypothesis (Theorem 3) by A. Akbary and Z. Friggstad, published in American Mathematical Monthly, March 2009. /Pontus (talk) 16:47, 14 April 2009 (UTC)

"Probably true"

The statement in the lead section that the Riemann Hypothesis is "probably true" is a logical faux pas, although it does echo the words of Riemann himself, who considered it "sehr wahrscheinlich" ("very likely") to be true. Since there are an infinite number of Riemann zeros, proving that ten trillion of them are on the critical line does not prove that they are all on the critical line, any more than rain today proves that there will be rain tomorrow. All it proves is that we have more computer power than when the hypothesis was proposed in 1859.--♦IanMacM♦ ^{(talk to me)} 11:02, 7 July 2009 (UTC)

The phrase "probably true" has been removed from the lead, since it is at best misleading and at worst wrong. A better explanation is given here, in which Keith Devlin states:

For instance, the first published calculation of zeros of the Riemann Zeta function dates back to 1903, when J.P. Gram computed the first 15 zeros (with imaginary part less than 50). Today, we know that the Riemann Hypothesis is true for the first ten trillion zeros. While these computations do not prove the hypothesis, they constitute information about it. In particular, they give us a measure of confidence in results proved under the assumption of RH.

Computational power should not be confused with probablility, since there are infinitely many Riemann zeroes and the problem would defeat heuristic justification.--♦IanMacM♦ ^{(talk to me)} 13:02, 25 September 2009 (UTC)

Your changes seem to consist of two things:

1. The removal of the "in mathematics" boilerplate at the start that is used on many mathematical articles in Wikipedia to alert the reader to the general context of the article's subject, and
2. The removal from the lede of any discussion of the significance or difficulty of the problem.

I disagree that these are constructive changes. —David Eppstein (talk) 13:06, 25 September 2009 (UTC)

Hopefully a reader would realize that the article was about mathematics even without the use of a boilerplate template (it does use rather a lot of mathematical language). Anyway, the main point was to point out that the 10 trillion figure has no basis in probability theory.--♦IanMacM♦ ^{(talk to me)} 13:21, 25 September 2009 (UTC)

Actually, it can be shown that it is probably true in a certain probabilistic sense. Brian Conrey has shown that at least 40% of the zeros lie on the critical line (before Conrey, N. Levinson and A. Selberg proved similar theorems with a smaller percentage). —Preceding unsigned comment added by 74.74.172.157 (talk • contribs)

Ingham

In the section "Distribution of prime numbers" there is the reference "(Ingham 1990)". I guess Ingham's book was reprinted in 1990, but it's ridiculous to refer to it that way.--Sandrobt (talk) 23:51, 8 September 2009 (UTC)

 Done —David Eppstein (talk) 01:00, 9 September 2009 (UTC)

Protection

Leaving aside the vandalism that abuses some of the formulae, there seems to be an editor or faction that imagines that a solution has been found to the RH, but does not cite any sources. As far as I'm concerned, if this has happened, it is a major breakthrough and would be widely reported. If so, such a solution requires rigorous peer-review before acceptance, and I have applied protection for an appropriate length to allow that review, acceptance, and promulgation, to occur. I have no doubt that should it happen, registered editors will be well-placed to edit this article; non-registered editors are otherwise invited to use {{editsemiprotected}} to place appropriate sources *here* first. P.S, anyone wanna buy some phlogiston? Rodhullandemu 00:36, 22 September 2009 (UTC)

Vasiliu Lucilius and Eugen Campu appear to be real people, see here and here. However, the claim to have proved Riemann in this edit is well into WP:REDFLAG territory. Who for instance, were the "over ten Fields Medal winners" who supported this claim? Do tell...--♦IanMacM♦ ^{(talk to me)} 14:15, 22 September 2009 (UTC).

Their signatures can be seen at www.petitiononline.com Math Cray

Forget it. This is the last of Hilbert's Problems and there is a prize of a million dollars. Any substantiated claim to a solution would make not only headlines in the mathematical journals, but also in headline, mainstream news. Unless and until this is reliably sourced, even by the mathematical community, signatures on an online petition are meaningless. If these Fields Medal winners collectively publish a paper authenticating this solution, fine, but until they, or others of equal authority do so, sorry, it's bollocks. Rodhullandemu 23:53, 22 September 2009 (UTC)

The reference given appears to be "Mathematical,Juridical,Political,Financial Help" at petitiononline dot com (Wikipedia automatically blocks links to this site). The signatures include this one by Jean Bourgain, who is indeed a Fields Medal winner:

I am Fields Medalist,professor at IAS Princeton and I fully sustain this petition,there are exceptional results ,specialy the proof of Riemann Hypothesis.

Barack Obama and Leon Panetta have also apparently signed this petition with the following:

The order of the President of U.S.A.and the Director of C.I.A.,Mr.Leon Panetta to the President of the Jassy Law Court and to,,Registratura,, womens is to take immediately the file 11.792/1991 in study from 1991 with the accusation of,,International terrorism,grave form,,.Any try of tergiversation of the judges and prosecutors from Jassy and Romania and all European Union and the ,,Registratura,, employees will be looked like a coalisation with this very dangerous terrorist group very well established and indicated by all the Secret Services of the World in the online petition.The victim has a very sophisticated microcip implanted in his optical system in the brain with which he films all the persons in front of him.All the images are transmited instantely at C.I.A.headquarter.Sincerely Barack Obama and Leon Panetta.

No comment...--♦IanMacM♦ ^{(talk to me)} 07:11, 23 September 2009 (UTC)

But have they obtained Riemann's signature yet? Surely that would lend more authority to their petition than these modern upstarts and politicians. —David Eppstein (talk) 12:41, 23 September 2009 (UTC)

Apparently Ban Ki-moon has also signed the petition, and has said "By the orders of all the governements of the world represented at United Nations we fully sustain the petition". Hut 8.5 13:09, 23 September 2009 (UTC)

Riemann Hypothesis Online Video Link

Hi,

I wanted to add an online video llink of the Riemann Hypothesis which explores background and gives a visual angle after the External Links Section. This gives more of a historical placing and visual overview for non-mathematicians. Would the people who are in charge here consider this link:

• The Riemann Hypothesis - A Visual Exploration — a visual exploration of Prime Numbers and the Riemann Hypothesis

If you have done this yourself, it should not have problems with the copyright (see WP:YOUTUBE). However, the video is impressionistic and does not add much to what the article already says (see WP:EL). The video is probably best approached through YouTube itself.--♦IanMacM♦ ^{(talk to me)} 12:12, 15 December 2009 (UTC)