Talk:Riemann hypothesis/Archive 3

Operator Theory
A new announced proof of RH using Operator theory (i.e te existence of a Hamiltonian having the energies satisfying $$ \zeta (1/2+iE_{n}) =0 $$ from the "Real sociedad Española de Matematicas" (Spanish royal society of Math)

Concrete Mathematics
In the answer to exercise 4.73 of their book "Concrete Mathematics", Graham, Knuth and Patashnik claim that the Riemann Hypothesis is equivalent to "the Zeta function is nonzero when the real part of its argument is greater than 1/2" (the usual formulation is with "greater than" replaced by "not". This is probably obvious but I don't see it right away. Can someone show me ?
 * For the critical strip it's the same. I don't know the book which is mentioned above, but with "not 1/2" is meant, that there is no zero for the riemann zetafunction in the critical strip beside on the critical line. "greater than 1/2" is more generally. We have -2n (n is a positive integer) as trivial nulls of the zetafunction, which are not considered, when "not 1/2" is used.

I understand. But is there an easy way to show that if there is no zero with real part > 1/2, then there is no nontrivial zero with real part != 1/2 ?
 * Yes, that's possible. With the functional equation of the Riemann Zetafunction. With real part <= 0 we have the trivial nulls, with real part >= 1 we have no nulls, and with the functional equation we have: If there is no zero with real part > 1/2 then we have no zero with real part < 1/2 and therefore no zero with real part != 1/2 . You find this equation for example with the link http://www.maths.ex.ac.uk/~mwatkins/zeta/fnleqn.htm but there are thousands of different scripts where you can find it too. (Anonymous conversation, circa 2-6 December 2005)


 * New discussion goes to the bottom of the page. Please sign and date your contributions. linas 13:49, 6 December 2005 (UTC)


 * Why don't we put new discussions at the top of the page, so that they stand out, letting the old, less active ones sediment at the bottom ?


 * Because the WP convention is that they go to the bottom. All the regulars expect to find the new stuff at the bottom. Another thing: it is customery to sign one's posts with four tildes, like so: ~ . linas 23:03, 12 December 2005 (UTC)

Is this a counterexample?
I'm not even really familiar with the Riemann hypothesis, but I stumbled upon this in the march 26 issue of postsecret: ''I willingly withheld the greatest discovery in mathematics from the world...It is the solution to the Riemann Hypothesis...The Riemman [sic] Hypothesis is false. The prime number 2^13466971-1 is off the critical line.'' Anyone confirm this as true or false?--Hearth 16:52, 26 March 2006 (UTC)


 * The number 2^13466971-1 is not prime, although it is certainly off the critical line. It is also clearly not a zero of the zeta function, for remember the definition of the zeta function: $$\zeta(s) = \sum_{n=1}^\infin \frac{1}{n^s}$$; if we plug in any positive integer (like the large number given) we will get some number greater than 1.--Rljacobson 03:01, 27 March 2006 (UTC)


 * Someone may want to look at the new "Solution" section that just popped up in the article; citing postsecret. I've emailed the blog maintainers in an attempt to clear this up.  --Ebrevdo 04:26, 27 March 2006 (UTC)

An answer?
Somebody posted the following [on http://postsecret.blogspot.com]:


 * "I willingly withheld the greatest discovery in mathematics from the world...It is the solution to the Riemann Hypothesis...The Riemman [sic] Hypothesis is false. The prime number 2^13466971-1 is

off the critical line."

I don't know exactly what to do with it, whether it warrants a notation here, but there it is. It'll be gone from its posting next Sunday, so if you want to check it out do it before then. Retinarow 07:28, 27 March 2006 (UTC)


 * Is that meant as a joke? I can't imagine why else someone would suggest that a posting at postsecret "warrants a notation here". -- Jibal 07:35, 10 May 2006 (UTC)


 * I'm convinced that the author on postsecret does not have a clue what he's talking about. -- Jitse Niesen (talk) 12:10, 27 March 2006 (UTC)


 * In particular, the "critical line" is the line Im(z)=.5, a line in the complex plane that every integer is off of. For example, 2^5-1 is off the critical line.   Don't stop the presses.  The Riemann hypothesis, as far as I understand it, isn't about prime numbers being on or off of any line.  The zeros of the zeta function, which are hypothesized to lie on the critical line, are distributed in a way related to the distribution of primes, but it isn't a simple one-to-one correspondence, so there's no reasonable way to talk about an individual prime number falling "off the line". -GTBacchus(talk) 16:31, 27 March 2006 (UTC)


 * No, the critical line is the line Re(z) = ½ .Daqu (talk) 07:49, 27 March 2008 (UTC)

Y'know, someone could be giving out the phone number (213) 466-9711 with this hoax. What's that area code, Los Angeles? -GTBacchus(talk) 20:42, 27 March 2006 (UTC)
 * That's so ridiculously unlikely that it has to be true. To the batcave, GTBacchus! Retinarow 20:56, 27 March 2006 (UTC)
 * Heyo. I read PS, so I thought to come here and see what was on the talk page about it...  I don't think it's a phone number.  I spent a little while on google using reverse phone lookup sites and every one of them so far says it's available/not a number in use.  Just food for thought.  -- Tabaqui 15:13, 30 March 2006 (UTC)

Hey, sorry for being stupid, but what does the -1 at the end of that number signify? Is it just one less than 2^13466971? 205.147.225.20 14:34, 29 March 2006 (UTC)


 * That's right. It's one less than a power of 2, which is a usual form for large primes.  Nothing to do with zeros of the zeta function, of course.  Of course, since 13466971 isn't prime (17 is a factor), then neither is 2^13466971-1 (2^17-1 is a factor). -GTBacchus(talk) 16:37, 29 March 2006 (UTC)


 * Good point, but 13466971 = 7 x 1923853, not 17. So two factors are (2^7-1) and (2^1923853-1). pstudier 22:49, 29 March 2006 (UTC)
 * Oops, I asked my calculator to factor the wrong number, obviously. Thanks for double-checking me. -GTBacchus(talk) 20:39, 30 March 2006 (UTC)

PLEASE: There is no need to commentate every nonsense which is written somewhere ! Aschauer 18:00 31 March 2006
 * There's no harm done, and possibly someone will learn something from this discussion. Of course, it has no place in the article, if that's what you meant... -GTBacchus(talk) 00:49, 1 April 2006 (UTC)
 * Yes, basically. Aschauer 3.April 2006

Since some people seem to actually believe that the Riemann Hypothesis has been disproved, maybe we should debunk this. The number is not prime, all positive integers are off the critical line, the number is not prime, the hypothesis is about zeros, not primes, etc. pstudier 20:57, 1 April 2006 (UTC)


 * I don't think that's a good idea. Do you want to debunk the many amateur proofs of difficult mathematical problems like Fermat's last theorem and the four colour theorem? That would give very long articles without much information and I think it would also run afoul of the no-original-research policy. I maintain that it's best not to mention non-newsworthy proofs at all. If necessary, we have tools to force this if this is what we want. -- Jitse Niesen (talk) 04:38, 3 April 2006 (UTC)


 * Uh, definitely not a good idea to be debunking it in the article though. Do you realize how many people at this very moment claim to have a proof or disproof of RH??  I know people in number theory and they're always getting emails, letters, etc. about this stuff; occasionally, in some crappy journal somewhere, a paper with a proof or disproof is published, when the editors fail to realize the main theorem actually implies or refutes RH (or apparently don't care).  Adding this stuff and then debunking it would add the equivalent of a few hundred pages to the article!  Mentioning famous claims, such as de Branges', is ok, but not this random crap.  --C S (Talk) 11:38, 10 April 2006 (UTC)

What I'm wondering is if the person who posted it means to say that it is 2^13466971-i or (2^13466971-1)i or (2^13466971)i-1 or something like that. --70.77.11.80 02:52, 12 May 2007 (UTC)

The critical line is Re(z) = .5, hence any permutation of the symbols with i's thrown in, are still not on the line. Furthermore, it is not our job to attempt to discover if some random person has made a typo.Phoenix1177 (talk) 05:18, 2 January 2008 (UTC)

In the News
New [|ARXIV] claims that Xian-Jin Li has a claimed proof of Riemann Hypothesis, as announced on Slashdot [].

Article was not cited but is a current event... and IANAM but looks promising. [| Seed Magazine] -- × × × jijin+machina | Chat Me! ×× × -- 16:17, 27 March 2006 (UTC)

Walts0042 Note about the above noted article (From the FEB/MAR 2006 issue of Seed)

[Prime Numbers Get Hitched] - In their search for patterns, mathematicians have uncovered unlikely connections between prime numbers and quantum physics. Will the subatomic world help reveal the elusive nature of the primes? by Marcus du Sautoy • Posted March 27, 2006 12:40 AM ''There is an important sequence of numbers called "the moments of the Riemann zeta function." Although we know abstractly how to define it, mathematicians have had great difficulty explicitly calculating the numbers in the sequence. We have known since the 1920s that the first two numbers are 1 and 2, but it wasn't until a few years ago that mathematicians conjectured that the third number in the sequence may be 42—a figure greatly significant to those well-versed in The Hitchhiker's Guide to the Galaxy.''

Walts0042 18:49, 30 March 2006 (UTC)


 * WP does not have, at this time, an article on the moments. It also doesn't have articles on GUE, GOE etc. The nuclear physics articles on WP basically completely suck. Actually, completely don't even exist. linas 20:11, 30 March 2006 (UTC)

How to prove the Riemann Hypothesis
My paper "how to prove the Riemann Hypothesis" was published in the web Journal"General Science Journal" on March 18th 2005.The address of the Journal is www.wbabin.net.It was published again in the Journal Spacetime&Substance,No.1,2006,P.1.Also it is  published in the Proceedings of PIRT-CMS-2007 Kolkata Fayez Fok Al Adeh

About the incomplete totality of the set of all prime natural numbers
Essay moved to User:BenCawaling/Essay. 08:43, 14 April 2006 (UTC)

Confusing definitions
There are some confusing definitions in this article: 1. The Riemann zeta function article says "The Riemann zeta-function ζ(s) is defined for any complex number s with real part > 1" but this article says "The Riemann zeta-function is defined for all complex numbers s ≠ 1" 2. This article says that there are zeros for s={-2,-4,-6...} but as in #1 there are no definition for the zeta-function for s <= 1. Also, replacing with s=-2 in the zeta-function does not result in zero..

Please, clarifications are necessary

Eledu 18:13, 15 April 2006 (UTC)


 * In response to (1), the Riemann zeta function article says that the Riemann zeta-function is defined for Re(s)>1 by the given summation formula, it goes on immediately to say that it can be extended by analytic continuation to the rest of the complex plane excluding s=1. When Re(s)<=1, the summation formula doesn't apply, but we're talking about the unique analytic extension of the function defined by that formula.


 * That answer sort of covers your second question as well. In Riemann zeta function, there are actually several methods given for extending the definition of the zeta function to all complex numbers (except for s=1).  How can we make that clearer in the original definition, do you think? -GTBacchus(talk) 18:29, 15 April 2006 (UTC)


 * It's perfectly clear as is. What isn't clear is why Eledu quoted only the first part of the first sentence and ignored the rest of the paragraph.  Saying that a function is defined for certain values by a specific equation is nothing like saying that the function is only defined for those values. -- Jibal 07:52, 10 May 2006 (UTC)

It's not defined from 0<Re(z)<1/2, but everything else is either obviously defined, or defined bby Ramanujan summation. -- He Who Is[ Talk ] 18:52, 13 July 2006 (UTC)

Unsolvability of the Riemann Hypothesis
Section moved to User:BenCawaling/Essay linas 13:15, 18 April 2006 (UTC)

What is a 'majority' of an infinite set?
''In 1914, Hardy proved that an infinite number of zeros lie on the critical line Re(s) = ½. However, it was still possible that an infinite number (and possibly the majority) of non-trivial zeros could lie elsewhere in the critical strip.''

Is it possible to clarify what is meant by 'the majority' here? Given that we're comparing two potentially-infinite sets here, it's not obvious to me whether the intended meaning here is something like "aleph-1 vs aleph-0", "there exists constant a such that for any finite x greater than a, the majority of the non-trivial zeros that have magnitude < a lie off the critical line", or something else. (I'm guessing it's not the first one, but my mathematics is a little rusty these days.) --Calair 06:35, 20 April 2006 (UTC)


 * I think, e.g. with "the majority of the zeroes lies on the critical line" is meant: If we have the first n zeroes, then exists a number N, so that for any n>N we have (number of all the zeroes off the line of the n zeroes)/(number of all the zeroes on the line of the n zeroes) < 50% . ... Aschauer 20.April 2006


 * But with infinite sets, that sort of statement is only meaningful when attached to a specific ordering of the zeroes. If there are infinitely many roots both on & off the critical line, it's possible to make that ratio come out to anything between 0 and 1 by ordering them accordingly.


 * For instance, suppose L1, L2, ... (bolded to make the next part more obvious) are the roots that lie on the critical line (it doesn't matter how we choose that particular ordering within these roots) and S1, S2, ... are the roots that lie elsewhere in the critical strip (again, ordering scheme within this subset of roots is unimportant). (BTW, I'm assuming that both these sets are aleph-null, or things get messier.)


 * Now suppose we order the complete set of roots thus:


 * L1, S1, L2, L3, S2, L4, L5, L6, S4, L7, L8, L9, L10, S5, ...


 * It can easily be seen that as n tends to infinity, the ratio of non-critical-line roots to total roots in the first n of the sequence converges to zero - even though all the non-critical-line roots are listed. But by exchanging the Ls and Ss in that sequence, I could just as easily make the ratio come out to 1.


 * There are some relatively obvious ways one might choose to order the roots - for instance, the second option in my previous comment is basically the same sort of statement as yours, with roots ordered by magnitude. Alternately, one might order them first by the magnitude of their imaginary part and second by the magnitude of their real part (which is not necessarily equivalent to the previous scheme, if there are roots off the critical line... and possibly not even a valid ordering scheme, if it turns out that there are infinitely many roots with the same imaginary part). But it's not obvious to me which of these or other orderings is intended here... and if no particular ordering was, then the statement is meaningless and adds nothing to the article. --Calair 00:00, 21 April 2006 (UTC)


 * Generally speaking, when speaking about the "ordering" of the zeroes of a zeta function, the ordering is by absolute value of imaginary part (you can alternate between negative and positive imaginary part if you like, it doesn't matter because they are symmetrical). There can't be infinitely many zeroes for a given imaginary part (or even with imaginary parts lying in a fixed bounded interval) because the zeroes of an analytic function form a discrete set, and it is known that the only zeroes of the zeta function (apart from the "trivial" zeroes) lie within the strip 0 < x < 1. This ordering is also the one that should be taken to make various sums (such as the explicit formula for the prime number counting function) converge. Dmharvey 00:15, 21 April 2006 (UTC)


 * Does discrete necessarily imply a finite number within a bounded interval, though? For instance, going by the definition at Isolated point, I'd have thought the set {0, 0.9, 0.99, 0.999, ...} was discrete.


 * (On further thought, it seems to me that it shouldn't be possible for a nontrivial analytic function to have an infinite number of zeroes within a bounded space, which answers my previous consideration - but I'm not sure that discreteness of zeroes is what it takes to prove that.) --Calair 03:34, 21 April 2006 (UTC)


 * Any infinite subset of a closed and bounded set in the complex plane must have a limit point, or "cluster point". (This is essentially the Bolzano–Weierstrass theorem)  Since the zeta function is analytic, and therefore continuous, the limit point of a bunch of zeros would have to be another zero, which would then not be isolated.
 * To apply to your example - if a function f(x) equals zero when x is any number in the set {.9, .99, .999, etc}, and f is a continuous function, then it's also equal to zero at x=1. This is because continuous functions map convergent sequences to convergent sequences. -GTBacchus(talk) 03:51, 21 April 2006 (UTC)


 * Ah, now I get it. Should have been able to figure that out from Dmharvey's response above, but too much engineering has rotted my brain ;-) --Calair 05:20, 21 April 2006 (UTC)


 * It's enough to define a clear order, e.g. use for all the zeroes 's' the absolute value |s|. ... Aschauer 21.April 2006
 * But if you count only the real parts (equal 1/2, unequal 1/2) of the known non-trivial zeroes then it's also enough. ... Aschauer 23.April 2006

I suspect (but I have not searched the literature) that majority is too weak; an infinite number of zeros on the line is consistent with almost all zeroes being off the line. Septentrionalis 16:44, 22 July 2006 (UTC)


 * I believe Selberg proved something like at least 3/5 of the zeroes are on the line. Dmharvey 18:21, 22 July 2006 (UTC)

Equivalents
I believe Lagarias' equivalent harmonic form can be manipulated to requiring

$$e(AQ)<=e(Hn)*ln(Hn)$$

where  ln(sigma(n)) <= A*Q(n)+ B    with B<=ln(2),H(n)=harmonic # and Q(n)=harmonic(n/2) ie half-harmonic

strict proof requires A<=1.0 (B=ln(2)). Actual numerics appear to show zero-intercept slopes for HCN>=7207200 all<A, so within bounds--Billymac00 17:18, 22 June 2006 (UTC)

btw the earlier link given by Fayez Fok Al Adeh shows that he claims to have proven RH in the paper "...Abstract: I have already discovered a simple proof of the Riemann Hypothesis..."

Superbly dumb question
Umm.. I'm sorry if this is the wrong place for this, but an awnser to my question might ultimatly help non-mathematicians to understand. That being said, I'm about to head off to uni to do maths, and could've overlooked it?

How the hell can a product of '1/ 's ever be zero (for when the Zeta zeroes occur). Help!? &mdash;the preceding comment is by 172.207.186.222 - 23:00, 23 August 2006 (UTC): Please sign your posts!


 * It can't. See and  above. -- EJ 03:23, 24 August 2006 (UTC)

Let me be direct. The Euler product representation for Zeta(z) is valid only for Re(z) > 1, and indeed the Zeta function has no zeroes in this region.Daqu (talk) 19:58, 24 March 2008 (UTC)

Demonstration of Trivial and Non-Trivial Zero Calculations?
Hi,]] Is it possible to provide links to pages (or separate wikipedia articles) for the actual calculations of the first trivial 0 (for -2) and the first non-trivial zero (for 1/2 +- 14.134725)?

What I have been looking for is a "walk-through" of the step-by-step calculations (without skipping any steps) of these calculations - plugging in the actual numbers at each step.

On many web sites it is said that it is easy to demonstrate the calculation of the trivial zeros - but I have not been able to find any demonstrations. As far as the non-trivial zeros, it is said the Riemann worked on the first few by hand - but again, I have not found any.

Thanks in advance! 71.233.175.121 16:04, 17 September 2006 (UTC)George

Strong Riemann Hypothesis
Is anyone aware of the (so-called) "strong Riemann hypothesis" - which states that all the non-trivial zeros of the Riemann zeta function are simple? I mention this because - in spite of the phenomenon of Lehmer zeros - it seems likely ... and note also that, given that the trivial zeros are all simple, it might be restated as: _all_ zeros of the Riemann zeta function are simple! —The preceding unsigned comment was added by 86.6.13.136 (talk • contribs) 23:22, 18 September 2006 (UTC2)

Delisted GA
Delisted article on the grounds of WP:WIAGA criteria 2b (the citation of its sources using inline citations is required). Tarret 21:27, 19 September 2006 (UTC)

Hlbert-Polya Approach
I think that the most relevant method to prove RH is via HIlbert-Polya operator having all its 'Eigenvalues' to be the imaginary parts of a certain Self-adjoint operator, for example a Hamiltonian H, in case this H has a complex potential the RH is completeley false. —The preceding unsigned comment was added by 85.85.100.144 (talk) 09:48, 11 January 2007 (UTC).

Proof by Fayez Fok al-Adeh
(Updated 12-28, 2005) In 2005 a proof was published in the General Science Journal Proof http://www.wbabin.net/aladeh/riemann.pdf linked from http://www.wbabin.net/papers.htm referred by http://www.ascssf.org.sy/Riemann%20Hypothesis.htm (If this proof is considered "phony", and this paragraph is to be removed, please email subs2718 [at] yahoo (dot) com with an explanation, why it is not a real exact proof.) I'm having difficulty logging in and using the Wikipedia "talk" feature. Please do email me if you have reviewed the proof or if you have comments. A.C.G


 * Moved here from main page. Mrhawley 17:15, 28 December 2006 (UTC)


 * A similar proof involving Post inversion formula and Dirichlet series generating function can be found at http://www.wbabin.net/science/moreta3.pdf  the author gives a criterion using a limit to prove or disprove RH based on the growth for the Mertens function

How to Prove the Riemann Hypothesis:Fayez Fok Al Adeh
I have proved The Riemann Hypothesis.My proof is exact.My proof entitled"How to Prove the Rimann Hypothesis"was published in the web Journal"General Science Journal"on March 18th 2005.Thousands of mathematicians have read the proof through this Journal.The proof was published again in the Journal Spacetime&Substance,No 1,2006 P 1.Also it is published in the Proceedings of PIRT-CMS-2007 Kolkata Fayez Fok Al Adeh. —The preceding unsigned comment was added by 88.86.31.1 (talk) 07:09, 19 January 2007 (UTC).

Sorry, i wouldn't like to offend prof. Fayez, however i believe that the representation of the Riemann Zeta function in the form:

$$ \zeta(s)=s\int_{0}^{\infty}([x]-x)x^{-s-1} $$

is not valid for 1>s>0, the critical strip where all NOn-trivial zeros lie. —Preceding unsigned comment added by Karl-H (talk • contribs)


 * The representation of The Riemann Zeta Function mentioned in the above paragraph is the correct one.Please refer to any textbook on the subject.s is complex. —Preceding unsigned comment added by 213.178.224.163 (talk)


 * According to this page, the closest representation to that quoted above is $$\zeta(s)=\frac{1}{s-1}+1-s\int_1^\infty \frac{x-[x]}{x^{s+1}}dx \qquad\sigma>0$$. Seems to me that the $$\frac{1}{s-1}+1$$ bit is rather important when you're looking for zeroes... Gmalivuk 16:24, 21 March 2007 (UTC)

I would like to offend Fayez. If the Clay Institute hasn't given your million yet, shut the fuck up (with apologies to the rest of wikipedia). Also: can anyone explain to me what the external link is doing at the bottom of the page? I started reading it and it seems almost as crank-y as Fayez. Maybe it gets around to talking about Riemann after enough rants on other subjects, but I'm going to remove it unless someone can verify that it actually has substance. Tenebrous 03:31, 31 January 2007 (UTC)


 * As far as I can see, the Riemann hypothesis is not discussed there at all. I could only see references to Riemannian geometry. So I removed the link. -- Jitse Niesen (talk) 05:01, 31 January 2007 (UTC)

For those who are interested, here's a link to the "proof". CloudNine 18:40, 17 February 2007 (UTC)

This "proof" is in any case invalid. The variable x on the LHS and the variable x on the RHS in eq (24) are not the same (see (18) and (22)).

Misc
Historically, Riemann himself stated the conjecture in terms of his xi function

http://www.claymath.org/millennium/Riemann_Hypothesis/1859_manuscript/EZeta.pdf

"...it is very probable that all roots [of xi] are real..." rather than in terms of zeta.

There are explicit formulas for zeta inside the critical strip; here are 3 with proofs of 2 of them:

http://planetmath.org/?op=getobj&from=objects&id=4040

I would include one of these formulas early in the article, to help avoid the "traditional" confusion that results from stating the conjecture in terms of an analytic continuation.


 * Er, on second thought, those formulas appear in the Wiki article about the zeta function itself. Not much reason to repeat them. LDH 04:01, 25 February 2007 (UTC)

It is conjectured that all the nontrivial zeros of zeta and xi are simple zeros.

LDH 04:01, 25 February 2007 (UTC)

Hilbert-Polya conjecture
I removed the following text form the Hilbert-Polya section, because I cannot make sense of it:


 * Amazingly using V. Mangoldt formula and Semiclassical approach we reach to the conclusion that if H (Hermitian operator) exists then :


 * 1) $$ \sum_{n}e^{iu E_{n}}=Z(u)=e^{u/2}-e^{-u/2} \frac{d\psi _{0}}{du}-\frac{e^{u/2}}{e^{3u}-e^{u}}=Tr(e^{iu\hat H }) $$


 * (Trace formula for the exponential of H )


 * 2) Iff H is a Hamiltonian so H=T+V then the potential Must satisfy the constraint:''


 * $$ \frac{Z(u)u^{1/2}}{\sqrt \pi }\sim \int_{-\infty}^{\infty}dxe^{i (uV(x)+ \pi /4 )} $$


 * for further reference see Chebyshev function article, and the section dedicated to Riemann Hypothesis.

linas 00:30, 19 February 2007 (UTC)

Exchange amateurs/professionals
After a recent change the article now reads "Unlike some other celebrated problems, it is more attractive to amateurs in the field than to professionals." I think the original version (with amateurs and professionals exchanged) makes more sense: it is the "simpler" problems like e.g. the Goldbach conjecture or Fermat's theorem which attract amateurs (which I read to mean non-mathematicians) while the technical nature of the Riemann hypothesis makes it considerably less attractive for laymen then for people working professionally in the field. So I would either switch the sentence back, or remove it completely. &mdash; Tobias Bergemann 08:51, 19 February 2007 (UTC)


 * I agree. The thing is certainly highly interesting to professionals; but merely stating the problem entails rather more technicalities than are seen in Goldbach, Collatz, etc. which get a lot of attention from amateurs. I would omit the professional/amateur remark, personally, since after all the distinction between the people is of no importance to the science of it all. LDH 03:59, 25 February 2007 (UTC)


 * The main problem it makes RH less atractive to 'amateurs' or pseudo-mathematician (although i don't agree with this insulting assertion) is that to 'solve' or at least try to solve anything you must be familiar to concepts involving complex analysis, which is far away from the High-School math they use to solve --Karl-H 15:05, 18 March 2007 (UTC)

A Disproof?
I recently came across this article, which claims to be a disproof of the Riemann Hypothesis. Unfortunately I don't remember enough of what I learned at school to be able to really evaluate it, but I suspect that since it hasn't made headlines (to my knowledge), it doesn't work. However, unlike certain proof and disproof "arguments" (such as that by the good Prof. Fayez), this one seems to have been done by competent mathematicians. So, is there any validity to it? Gmalivuk 16:18, 21 March 2007 (UTC)


 * Heard about it and am guessing just very, very off-top-of-head that it's being looked over/double-triple-etc.-checked perhaps? Schissel | Sound the Note! 19:04, 28 March 2007 (UTC)

PLEASE ONCE AND FOR ALL
Someone please explain how plugging in -2 gives a zero output. —The preceding unsigned comment was added by 72.231.153.109 (talk • contribs) 17:33, 25 March 2007 (UTC)


 * Frome the functional equation:



\zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s) $$

If you put -2n n=1,2,3,4,5,6,.... the sine gives a zero value, except for n=0 since at s=1 the Riemann zeta function has a pole —The preceding unsigned comment was added by 85.85.100.144 (talk • contribs) 19:06, 29 March 2007 (UTC)

Work of Montgomery
I just saw BBC TV program about the primes, which discussed RH in some detail. They went into details on work of Montgomery and others looking at the distribution of the zero which seemed to have a spectra similar to thouse found in random matrix theory. The Seed Magazine link has some more about this. Should this be addressed here? --Salix alba (talk) 00:55, 26 July 2007 (UTC)


 * Montgomery and he connection with random matrices is mentioned in the subarticle Hilbert-Pólya conjecture but indeed not in this article. I'm a bit surprised that it isn't mentioned, but I don't know the topic. -- Jitse Niesen (talk) 03:22, 26 July 2007 (UTC)

Helge von Koch and history of equivalent formuations.
WAREL has made an edit which constitutes a significant change in the history of equivalents, but I don't know if he's correct. Could someone research it? &mdash; Arthur Rubin | (talk) 15:14, 29 August 2007 (UTC)
 * Checked. -- Jitse Niesen (talk) 04:50, 30 August 2007 (UTC)

Cramer
the result "On the RH, the difference between the prime numbers is O(\squart{x)ln x)" is not from Cramer but from Von Koch.88.162.110.102 18:33, 5 November 2007 (UTC)

Simple zeros
It has been conjectured that all the zeros of zeta on the critical line are simple zeros, but the history of that claim, and the evidence in favour of it, are rather outside my competence. 209.121.88.198 (talk) 11:01, 9 March 2008 (UTC)

Erroneously captioned graph?
One of the very nice graphics accompanying this article is described as the real part of ζ(1/2 + it) versus the imaginary part of ζ(1/2 + it) for 0 ≤ t ≤ 34. Its caption, however, states: "This image shows a polar graph of the Riemann zeta function along the critical line."

The image in question is at http://en.wikipedia.org/wiki/Image:Zeta_polar.svg. Even the name "Zeta_polar" is misleading; in fact, I would call it "wrong".

I do not see anything "polar" about it. In fact, it seems confusing to say (with so many unnecessary words) that this is the real part of ζ(1/2 + it) versus the imaginary part of ζ(1/2 + it).

I recommend that this graphic should just be described as displaying the "values of ζ(1/2 + it) in the complex plane for 0 ≤ t ≤ 34". No mention of real or imaginary parts, and certainly not the word "polar", is useful here.Daqu (talk) 19:55, 24 March 2008 (UTC)

Wrong lower limit of integration?
In the section "The Riemann hypothesis and primes" of the article, the lower limit of integration in each of the two integrals is 0. These integrals are not convergent. Shouldn't the lower limit of integration in both cases be 2 ? Daqu (talk) 07:46, 27 March 2008 (UTC)

P.S. I now realize that if the principal value is calculated, then this integral (of 1/ln(t) from t = 0 to t = x) ought to converge. Perhaps for those among us who are non-experts in analytic number theory, this might be mentioned?Daqu (talk) 06:54, 2 April 2008 (UTC)

Fractal dimension
<< The self-similarity of the zero distributions is quite remarkable, and is characterized by a large fractal dimension of 1.9. >>

This sentence looks weird, but I am not a specialist.

1. If the fractal dimension were 1.9, then non-trivial zeros could not been all on the critical line. There is at list a typo in this number 1.9.

2. Moreover, is it a mathematical result ? Or is it based on numerical simulation ? Or is it only non rigorous mathematics ? This should be precised.


 * You should read the paper for definite answers. I had a brief look for you, and from that I would say:
 * 1. The number 1.9 is the (box) dimension of the graph of the distance between two consecutive zeros plotted against the imaginary part of the zeros.
 * 2. The result is derived from a statistical analysis of a number (I think ten thousands) of zeros that are computed numerically.
 * I agree that this should be clarified if the sentence is to stay. However, I'm wondering whether it would be better to delete this result because it is not connected to the rest of the article. -- Jitse Niesen (talk) 12:15, 8 July 2008 (UTC)

OK, thanks. I deleted the statement but I think that it could be reinserted if precisely enough stated. —Preceding unsigned comment added by 88.178.225.58 (talk) 11:56, 9 July 2008 (UTC)

What does "giving" a theorem mean?
At Riemann_hypothesis, the article says "[...] Robin's theorem [...] was given by Guy Robin in 1984." What does "giving" a theorem mean? Is it like proving it? or suggesting, or conjecturing? If so, then some other wording might be better. --Mike Schwartz (talk) 21:25, 29 September 2008 (UTC)

A few roots?
I think the article should include some of the non-trivial roots, to some approximation. Maybe a couple of small roots, and a couple of big roots. Maybe a graph showing the distribution of them. Albmont (talk) 12:23, 15 October 2008 (UTC)