Talk:Riemann hypothesis/Archive 3

2386:" 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. — 31: 2428:, 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? 995:

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

892:, ... (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 973:

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

798:

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. --

684:

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,

1859:

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

2408:

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

972:

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

163:

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".

2396:

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

1508:

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

994:

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

962:

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

1369:

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

851:

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

168:

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

594:

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

477:

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. --

2200: 776:, 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. 874:

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

2755:

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."

1350:

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.

2637:

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

569:

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

1082:

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.

1419:

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.

558:

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.

420:, 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". - 502:

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).

682:- 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 2351: 2605: 1805: 595:

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. --

2792:

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 ?

846:

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

1045:

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.)

1273:

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?

751:

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..

1468:

with an explanation, why it is not a real exact proof.) I'm having difficulty logging in and using the Knowledge "talk" feature. Please do email me if you have reviewed the proof or if you have comments. A.C.G

331: 698:

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.

785:, 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? - 250:

I willingly withheld the greatest discovery in mathematics from the world...It is the solution to the Riemann Hypothesis...The Riemman Hypothesis is false. The prime number 2^13466971-1 is off the critical

685:

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.

1975: 617:

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

1621: 226:

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: ~~~~.

2853:

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. --

2925:

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.

1344:

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)?

416:

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

2766:

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).

366:"I willingly withheld the greatest discovery in mathematics from the world...It is the solution to the Riemann Hypothesis...The Riemman Hypothesis is false. The prime number 2^13466971-1 is 2736:

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.

373:

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.

263:

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:

2811:

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?

1265:

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..."

1248: 1925:

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.

151: 1347:

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.

347:

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. --

1847: 2226: 2487: 1677: 1079:) 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. 718:.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 1860:

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.

866:

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: -->

1650:

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.

543:

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... -

867:

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

2906:

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. --

714:

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

1870:

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. --

977:

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. --

608:

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. --

2840:

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.

178:

Yes, that's possible. With the functional equation of the Riemann Zetafunction. With real part <= 0 we have the trivial nulls, with real part : -->

2847:

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.

2837:

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.

1512: 1423: 1373: 1297: 1140:

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

574:. 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. -- 180:

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

1898:

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)).

1657: 2830:<< The self-similarity of the zero distributions is quite remarkable, and is characterized by a large fractal dimension of 1.9. : --> 2737: 2472: 211:

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 ?

2875: 2625: 1427: 2688:

which constitutes a significant change in the history of equivalents, but I don't know if he's correct. Could someone research it? —

2195:{\displaystyle \sum _{n}e^{iuE_{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}}})} 1964:

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

266: 383:

Is that meant as a joke? I can't imagine why else someone would suggest that a posting at postsecret "warrants a notation here". --

1385: 81: 76: 64: 59: 1531:

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

2610:

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

1929:

Er, on second thought, those formulas appear in the Wiki article about the zeta function itself. Not much reason to repeat them.

1516: 812:

It's not defined from 0<Re(z)<1/2, but everything else is either obviously defined, or defined bby Ramanujan summation. --

2850:

2. The result is derived from a statistical analysis of a number (I think ten thousands) of zeros that are computed numerically.

1076: 2438:

Heard about it and am guessing just very, very off-top-of-head that it's being looked over/double-triple-etc.-checked perhaps?

1536: 852:

majority of the non-trivial zeros that have magnitude < a lie off the critical line", or something else. (I'm guessing it's

1509:

Spacetime&Substance,No 1,2006 P 1.Also it is published in the Proceedings of PIRT-CMS-2007 Kolkata Fayez Fok Al Adeh.

1020:

Does discrete necessarily imply a finite number within a bounded interval, though? For instance, going by the definition at

2722:

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.

492:

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?

456:

Y'know, someone could be giving out the phone number (213) 466-9711 with this hoax. What's that area code, Los Angeles? -

1642: 1156: 860: 248:

I'm not even really familiar with the Riemann hypothesis, but I stumbled upon this in the march 26 issue of postsecret:

1137:

It's enough to define a clear order, e.g. use for all the zeroes 's' the absolute value |s|. ... Aschauer 21.April 2006

2658: 94:

A new announced proof of RH using Operator theory (i.e te existence of a Hamiltonian having the energies satisfying

38: 1909: 1661: 1252:

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

539:

PLEASE: There is no need to commentate every nonsense which is written somewhere ! Aschauer 18:00 31 March 2006

2468: 2741: 2879: 1255:

strict proof requires A<=1.0 (B=ln(2)). Actual numerics appear to show zero-intercept slopes for HCN: -->

2911: 2661:

but indeed not in this article. I'm a bit surprised that it isn't mentioned, but I don't know the topic. --

2621: 2384:

Unlike some other celebrated problems, it is more attractive to amateurs in the field than to professionals.

1431: 333:; if we plug in any positive integer (like the large number given) we will get some number greater than 1.-- 47: 17: 2387: 1381: 1182: 2858: 2707: 2693: 2666: 1875: 1799:{\displaystyle \zeta (s)={\frac {1}{s-1}}+1-s\int _{1}^{\infty }{\frac {x-}{x^{s+1}}}dx\qquad \sigma : --> 1485: 1288: 828: 728: 658: 579: 399: 2868:

OK, thanks. I deleted the statement but I think that it could be reinserted if precisely enough stated.

2723: 2639: 2464: 1355: 590:

Uh, definitely not a good idea to be debunking it in the article though. Do you realize how many people

493: 97: 2617: 782: 768: 744: 623: 179:= 1 we have no nulls, and with the functional equation we have: If there is no zero with real part : --> 2346:{\displaystyle {\frac {Z(u)u^{1/2}}{\sqrt {\pi }}}\sim \int _{-\infty }^{\infty }dxe^{i(uV(x)+\pi /4)}} 1811: 1444: 2844:

You should read the paper for definite answers. I had a brief look for you, and from that I would say:

1377: 619: 609: 2871: 2647: 2613: 2460: 1917:

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

1653: 1630: 1520: 747:

article says "The Riemann zeta-function ζ(s) is defined for any complex number s with real part : -->

2600:{\displaystyle \zeta (s)=2^{s}\pi ^{s-1}\sin \left({\frac {\pi s}{2}}\right)\Gamma (1-s)\zeta (1-s)} 525:

Oops, I asked my calculator to factor the wrong number, obviously. Thanks for double-checking me. -

479: 2907: 2895: 2443: 2410: 2361: 1920: 1671: 1634: 1959:

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

758: 2773:". No mention of real or imaginary parts, and certainly not the word "polar", is useful here. 1496:

the author gives a criterion using a limit to prove or disprove RH based on the growth for the

516:

Good point, but 13466971 = 7 x 1923853, not 17. So two factors are (2^7-1) and (2^1923853-1).

2930: 2854: 2703: 2689: 2662: 1871: 748:

1" but this article says "The Riemann zeta-function is defined for all complex numbers s ≠ 1"

654: 639:] claims that Xian-Jin Li has a claimed proof of Riemann Hypothesis, as announced on Slashdot 575: 395: 1324:

Let me be direct. The Euler product representation for Zeta(z) is valid only for Re(z) : -->

394:

I'm convinced that the author on postsecret does not have a clue what he's talking about. --

2899: 1638: 1497: 1489: 1450:(Updated 12-28, 2005) In 2005 a proof was published in the General Science Journal Proof 996: 173:

I understand. But is there an easy way to show that if there is no zero with real part : -->

2934: 2915: 2883: 2862: 2820: 2802: 2782: 2745: 2726: 2711: 2696: 2670: 2651: 2643: 2446: 2432: 2413: 2401: 2390: 2372: 1949: 1933: 1893: 1879: 1864: 1853: 1524: 1477: 1435: 1409: 1398: 1358: 1334: 1318: 1292: 1260: 1257: 1166: 1153: 1120: 1087: 1050: 1003: 981: 835: 816: 803: 789: 761: 703: 692: 671: 627: 612: 602: 583: 563: 547: 529: 520: 507: 496: 482: 472: 460: 446: 424: 412:

In particular, the "critical line" is the line Im(z)=.5, a line in the complex plane that

403: 387: 377: 351: 337: 334: 257: 231: 198: 1464:(If this proof is considered "phony", and this paragraph is to be removed, please email 194:

New discussion goes to the bottom of the page. Please sign and date your contributions.

2816: 2798: 2778: 2677: 1890: 1861: 1330: 1084: 1021: 813: 786: 689: 676: 544: 526: 504: 469: 457: 442: 421: 374: 1370:

simple, it might be restated as: _all_ zeros of the Riemann zeta function are simple!

181: 2439: 2429: 1850: 1474: 1163: 1113: 1000: 599: 571: 560: 517: 169:

trivial nulls of the zetafunction, which are not considered, when "not 1/2" is used .

1459: 1162:

I believe Selberg proved something like at least 3/5 of the zeroes are on the line.

2926: 1914:"...it is very probable that all roots are real..." rather than in terms of zeta. 644:

Article was not cited but is a current event... and IANAM but looks promising. --

468:

That's so ridiculously unlikely that it has to be true. To the batcave, GTBacchus!

348: 2215:

2) Iff H is a Hamiltonian so H=T+V then the potential Must satisfy the constraint:

2657:

Montgomery and he connection with random matrices is mentioned in the subarticle

1071:

Any infinite subset of a closed and bounded set in the complex plane must have a

153:

from the "Real sociedad Española de Matematicas" (Spanish royal society of Math)

1906:

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

1406: 1117: 1072: 1047: 978: 857: 254: 46:

If you wish to start a new discussion or revive an old one, please do so on the

2763:. Even the name "Zeta_polar" is misleading; in fact, I would call it "wrong". 1910:

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

2369: 1149: 832: 800: 700: 384: 228: 195: 164:

This is probably obvious but I don't see it right away. Can someone show me ?

2397:

distinction between the people is of no importance to the science of it all.

1177:

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

2903: 2812: 2794: 2774: 1493: 1326: 438: 1886: 1451: 2769:

I recommend that this graphic should just be described as displaying the "

679:

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

2398: 1946: 1930: 596: 2760: 1148:

is too weak; an infinite number of zeros on the line is consistent with

184:

but there are thousands of different scripts where you can find it too.

2425: 1315: 154: 1116:'s response above, but too much engineering has rotted my brain ;-) -- 771:

article says that the Riemann zeta-function is defined for Re(s): -->

722:

About the incomplete totality of the set of all prime natural numbers

1939:

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

1455: 1024:, I'd have thought the set {0, 0.9, 0.99, 0.999, ...} was discrete. 856:

the first one, but my mathematics is a little rusty these days.) --

326:{\displaystyle \zeta (s)=\sum _{n=1}^{\infty }{\frac {1}{n^{s}}}} 1112:

Ah, now I get it. Should have been able to figure that out from

2457:

Someone please explain how plugging in -2 gives a zero output.

174:

1/2, then there is no nontrivial zero with real part != 1/2 ?

25: 1921:

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

1325:

1, and indeed the Zeta function has no zeroes in this region.

781:

That answer sort of covers your second question as well. In

2896:

Riemann_hypothesis#Growth_rates_of_multiplicative_functions

2642:

has some more about this. Should this be addressed here? --

1340:

Demonstration of Trivial and Non-Trivial Zero Calculations?

1849:

bit is rather important when you're looking for zeroes...

1616:{\displaystyle \zeta (s)=s\int _{0}^{\infty }(-x)x^{-s-1}} 2771:

values of ζ(1/2 + it) in the complex plane for 0 ≤ t ≤ 34

2364:

article, and the section dedicated to Riemann Hypothesis.

1282:'s ever be zero (for when the Zeta zeroes occur). Help!? 1144:

I suspect (but I have not searched the literature) that

715: 2834:

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

2685: 1627:

0, the critical strip where all NOn-trivial zeros lie.

2490: 2229: 1978: 1814: 1681: 1674:, the closest representation to that quoted above is 1539: 1504:

How to Prove the Riemann Hypothesis:Fayez Fok Al Adeh

1185: 740:

There are some confusing definitions in this article:

269: 100: 905:

Now suppose we order the complete set of roots thus:

182:

http://www.maths.ex.ac.uk/~mwatkins/zeta/fnleqn.htm

2599: 2345: 2194: 1841: 1798: 1615: 1242: 999:for the prime number counting function) converge. 325: 145: 1460:http://www.ascssf.org.sy/Riemann%20Hypothesis.htm 186:(Anonymous conversation, circa 2-6 December 2005) 1075:, or "cluster point". (This is essentially the 2382:After a recent change the article now reads " 437:No, the critical line is the line Re(z) = ½ . 8: 2761:http://en.wikipedia.org/Image:Zeta_polar.svg 1041:(On further thought, it seems to me that it 840: 1311: 1307: 2542: 2520: 2510: 2489: 2330: 2302: 2286: 2278: 2253: 2249: 2230: 2228: 2209:(Trace formula for the exponential of H ) 2176: 2175: 2168: 2143: 2127: 2112: 2108: 2102: 2082: 2072: 2062: 2055: 2038: 2034: 2004: 1993: 1983: 1977: 1815: 1813: 1765: 1742: 1736: 1731: 1697: 1680: 1598: 1567: 1562: 1538: 1494:http://www.wbabin.net/science/moreta3.pdf 1401:criteria 2b (the citation of its sources 1184: 315: 306: 300: 289: 268: 128: 110: 99: 1452:http://www.wbabin.net/aladeh/riemann.pdf 841:What is a 'majority' of an infinite set? 822:Unsolvability of the Riemann Hypothesis 253:Anyone confirm this as true or false?-- 2680:and history of equivalent formuations. 1281:can a product of '1/<something: --> 1243:{\displaystyle e(AQ)<=e(Hn)*ln(Hn)} 44:Do not edit the contents of this page. 1256:=7207200 all<A, so within bounds-- 755:Please, clarifications are necessary 552:Yes, basically. Aschauer 3.April 2006 7: 1492:generating function can be found at 146:{\displaystyle \zeta (1/2+iE_{n})=0} 1397:Delisted article on the grounds of 896:aleph-null, or things get messier.) 710:How to prove the Riemann Hypothesis 2890:What does "giving" a theorem mean? 2561: 2287: 2282: 1842:{\displaystyle {\frac {1}{s-1}}+1} 1737: 1568: 1403:using inline citations is required 1388:) 23:22, 18 September 2006 (UTC2) 301: 24: 2788:Wrong lower limit of integration? 2807:P.S. I now realize that if the 1456:http://www.wbabin.net/papers.htm 362:Somebody posted the following : 29: 2378:Exchange amateurs/professionals 667: 664: 653: 648: 645: 2916:21:25, 29 September 2008 (UTC) 2594: 2582: 2576: 2564: 2500: 2494: 2480:Frome the functional equation: 2338: 2321: 2315: 2306: 2242: 2236: 2189: 2181: 2161: 2024: 2018: 1885:For those who are interested, 1757: 1751: 1691: 1685: 1591: 1582: 1576: 1573: 1549: 1543: 1410:21:27, 19 September 2006 (UTC) 1359:16:04, 17 September 2006 (UTC) 1237: 1228: 1216: 1207: 1198: 1189: 774:by the given summation formula 279: 273: 134: 104: 1: 2628:) 19:06, 29 March 2007 (UTC) 2475:) 17:33, 25 March 2007 (UTC) 2402:03:59, 25 February 2007 (UTC) 2391:08:51, 19 February 2007 (UTC) 2373:00:30, 19 February 2007 (UTC) 1950:04:01, 25 February 2007 (UTC) 1934:04:01, 25 February 2007 (UTC) 1894:18:40, 17 February 2007 (UTC) 1785: 1478:17:15, 28 December 2006 (UTC) 732:. 08:43, 14 April 2006 (UTC) 232:23:03, 12 December 2005 (UTC) 2935:12:23, 15 October 2008 (UTC) 2759:The image in question is at 2751:Erroneously captioned graph? 2727:18:33, 5 November 2007 (UTC) 1887:here's a link to the "proof" 1880:05:01, 31 January 2007 (UTC) 1865:03:31, 31 January 2007 (UTC) 1525:07:09, 19 January 2007 (UTC) 1436:09:48, 11 January 2007 (UTC) 199:13:49, 6 December 2005 (UTC) 2712:04:50, 30 August 2007 (UTC) 2697:15:14, 29 August 2007 (UTC) 1473:Moved here from main page. 1319:03:23, 24 August 2006 (UTC) 1293:23:00, 23 August 2006 (UTC) 1286:the preceding comment is by 1152:zeroes being off the line. 1077:Bolzano–Weierstrass theorem 628:05:18, 2 January 2008 (UTC) 572:no-original-research policy 2950: 2803:07:46, 27 March 2008 (UTC) 2783:19:55, 24 March 2008 (UTC) 2447:19:04, 28 March 2007 (UTC) 2433:16:18, 21 March 2007 (UTC) 2414:15:05, 18 March 2007 (UTC) 2360:for further reference see 1854:16:24, 21 March 2007 (UTC) 1484:A similar proof involving 1335:19:58, 24 March 2008 (UTC) 1121:05:20, 21 April 2006 (UTC) 1088:03:51, 21 April 2006 (UTC) 1051:03:34, 21 April 2006 (UTC) 1004:00:15, 21 April 2006 (UTC) 982:00:00, 21 April 2006 (UTC) 861:06:35, 20 April 2006 (UTC) 836:13:15, 18 April 2006 (UTC) 790:18:29, 15 April 2006 (UTC) 762:18:13, 15 April 2006 (UTC) 704:20:11, 30 March 2006 (UTC) 693:18:49, 30 March 2006 (UTC) 672:16:17, 27 March 2006 (UTC) 603:11:38, 10 April 2006 (UTC) 530:20:39, 30 March 2006 (UTC) 521:22:49, 29 March 2006 (UTC) 508:16:37, 29 March 2006 (UTC) 497:14:34, 29 March 2006 (UTC) 483:15:13, 30 March 2006 (UTC) 473:20:56, 27 March 2006 (UTC) 461:20:42, 27 March 2006 (UTC) 447:07:49, 27 March 2008 (UTC) 425:16:31, 27 March 2006 (UTC) 418:zeros of the zeta function 404:12:10, 27 March 2006 (UTC) 378:07:28, 27 March 2006 (UTC) 352:04:26, 27 March 2006 (UTC) 338:03:01, 27 March 2006 (UTC) 258:16:52, 26 March 2006 (UTC) 2821:06:54, 2 April 2008 (UTC) 2746:11:01, 9 March 2008 (UTC) 2671:03:22, 26 July 2007 (UTC) 2652:00:55, 26 July 2007 (UTC) 1466:subs2718 yahoo (dot) com 1365:Strong Riemann Hypothesis 1261:17:18, 22 June 2006 (UTC) 1167:18:21, 22 July 2006 (UTC) 1157:16:44, 22 July 2006 (UTC) 817:18:52, 13 July 2006 (UTC) 584:04:38, 3 April 2006 (UTC) 564:20:57, 1 April 2006 (UTC) 548:00:49, 1 April 2006 (UTC) 244:Is this a counterexample? 2884:11:56, 9 July 2008 (UTC) 2863:12:15, 8 July 2008 (UTC) 2659:Hilbert-Pólya conjecture 1955:Hilbert-Polya conjecture 1808:. Seems to me that the 804:07:52, 10 May 2006 (UTC) 767:In response to (1), the 613:02:52, 12 May 2007 (UTC) 388:07:35, 10 May 2006 (UTC) 370:off the critical line." 2453:PLEASE ONCE AND FOR ALL 2424:I recently came across 1625:is not valid for 1: --> 18:Talk:Riemann hypothesis 2898:, the article says " 2601: 2347: 2196: 1843: 1801: 1617: 1486:Post inversion formula 1312:#Confusing definitions 1308:#Analytic continuation 1298:Please sign your posts 1269:Superbly dumb question 1244: 884:For instance, suppose 829:User:BenCawaling/Essay 729:User:BenCawaling/Essay 327: 305: 147: 2616:comment was added by 2602: 2463:comment was added by 2348: 2197: 1844: 1802: 1618: 1515:comment was added by 1426:comment was added by 1415:Hlbert-Polya Approach 1376:comment was added by 1245: 783:Riemann zeta function 769:Riemann zeta function 745:Riemann zeta function 736:Confusing definitions 328: 285: 148: 42:of past discussions. 2488: 2227: 1976: 1812: 1679: 1537: 1183: 267: 159:Concrete Mathematics 98: 2291: 1741: 1572: 1354:Thanks in advance! 592:at this very moment 2684:WAREL has made an 2633:Work of Montgomery 2597: 2362:Chebyshev function 2343: 2274: 2192: 1988: 1839: 1796: 1786: 1727: 1613: 1558: 1240: 323: 143: 2886: 2874:comment added by 2826:Fractal dimension 2629: 2555: 2476: 2269: 2268: 2184: 2150: 2097: 1979: 1831: 1777: 1713: 1665: 1656:comment added by 1646: 1633:comment added by 1528: 1445:Fayez Fok al-Adeh 1439: 1389: 1302: 827:Section moved to 662: 321: 87: 86: 54: 53: 48:current talk page 2941: 2869: 2611: 2606: 2604: 2603: 2598: 2560: 2556: 2551: 2543: 2531: 2530: 2515: 2514: 2458: 2388:Tobias Bergemann 2352: 2350: 2349: 2344: 2342: 2341: 2334: 2290: 2285: 2270: 2264: 2263: 2262: 2261: 2257: 2231: 2201: 2199: 2198: 2193: 2188: 2187: 2186: 2185: 2177: 2151: 2149: 2148: 2147: 2135: 2134: 2121: 2120: 2116: 2103: 2098: 2096: 2088: 2087: 2086: 2073: 2071: 2070: 2066: 2047: 2046: 2042: 2011: 2010: 2009: 2008: 1987: 1848: 1846: 1845: 1840: 1832: 1830: 1816: 1807: 1804: 1803: 1797: 1778: 1776: 1775: 1760: 1743: 1740: 1735: 1714: 1712: 1698: 1651: 1628: 1622: 1620: 1619: 1614: 1612: 1611: 1571: 1566: 1510: 1498:Mertens function 1490:Dirichlet series 1421: 1371: 1283: 1249: 1247: 1246: 1241: 997:explicit formula 669: 666: 661: 652: 650: 647: 332: 330: 329: 324: 322: 320: 319: 307: 304: 299: 152: 150: 149: 144: 133: 132: 114: 73: 56: 55: 33: 32: 26: 2949: 2948: 2944: 2943: 2942: 2940: 2939: 2938: 2923: 2900:Robin's theorem 2892: 2828: 2809:principal value 2790: 2753: 2734: 2720: 2682: 2635: 2612:—The preceding 2544: 2538: 2516: 2506: 2486: 2485: 2459:—The preceding 2455: 2444:Sound the Note! 2422: 2409:use to solve -- 2380: 2298: 2245: 2232: 2225: 2224: 2164: 2139: 2123: 2122: 2104: 2089: 2078: 2074: 2051: 2030: 2000: 1989: 1974: 1973: 1957: 1904: 1820: 1810: 1809: 1761: 1744: 1702: 1676: 1675: 1658:213.178.224.163 1594: 1535: 1534: 1511:—The preceding 1506: 1448: 1422:—The preceding 1417: 1395: 1372:—The preceding 1367: 1342: 1289:172.207.186.222 1271: 1253: 1250: 1181: 1180: 1175: 1154:Septentrionalis 843: 824: 738: 727:Essay moved to 724: 712: 635: 360: 311: 265: 264: 246: 161: 124: 96: 95: 92: 90:Operator Theory 69: 30: 22: 21: 20: 12: 11: 5: 2947: 2945: 2922: 2919: 2891: 2888: 2866: 2865: 2851: 2848: 2845: 2827: 2824: 2789: 2786: 2752: 2749: 2738:209.121.88.198 2733: 2730: 2724:88.162.110.102 2719: 2716: 2715: 2714: 2681: 2678:Helge von Koch 2675: 2674: 2673: 2634: 2631: 2608: 2607: 2596: 2593: 2590: 2587: 2584: 2581: 2578: 2575: 2572: 2569: 2566: 2563: 2559: 2554: 2550: 2547: 2541: 2537: 2534: 2529: 2526: 2523: 2519: 2513: 2509: 2505: 2502: 2499: 2496: 2493: 2482: 2481: 2465:72.231.153.109 2454: 2451: 2450: 2449: 2421: 2418: 2417: 2416: 2405: 2404: 2379: 2376: 2367: 2366: 2356: 2355: 2354: 2353: 2340: 2337: 2333: 2329: 2326: 2323: 2320: 2317: 2314: 2311: 2308: 2305: 2301: 2297: 2294: 2289: 2284: 2281: 2277: 2273: 2267: 2260: 2256: 2252: 2248: 2244: 2241: 2238: 2235: 2219: 2218: 2212: 2211: 2205: 2204: 2203: 2202: 2191: 2183: 2180: 2174: 2171: 2167: 2163: 2160: 2157: 2154: 2146: 2142: 2138: 2133: 2130: 2126: 2119: 2115: 2111: 2107: 2101: 2095: 2092: 2085: 2081: 2077: 2069: 2065: 2061: 2058: 2054: 2050: 2045: 2041: 2037: 2033: 2029: 2026: 2023: 2020: 2017: 2014: 2007: 2003: 1999: 1996: 1992: 1986: 1982: 1967: 1966: 1956: 1953: 1937: 1936: 1903: 1900: 1883: 1882: 1857: 1856: 1838: 1835: 1829: 1826: 1823: 1819: 1795: 1792: 1789: 1784: 1781: 1774: 1771: 1768: 1764: 1759: 1756: 1753: 1750: 1747: 1739: 1734: 1730: 1726: 1723: 1720: 1717: 1711: 1708: 1705: 1701: 1696: 1693: 1690: 1687: 1684: 1667: 1666: 1610: 1607: 1604: 1601: 1597: 1593: 1590: 1587: 1584: 1581: 1578: 1575: 1570: 1565: 1561: 1557: 1554: 1551: 1548: 1545: 1542: 1505: 1502: 1501: 1500: 1481: 1480: 1447: 1441: 1416: 1413: 1394: 1391: 1366: 1363: 1356:71.233.175.121 1341: 1338: 1322: 1321: 1306:It can't. See 1270: 1267: 1251: 1239: 1236: 1233: 1230: 1227: 1224: 1221: 1218: 1215: 1212: 1209: 1206: 1203: 1200: 1197: 1194: 1191: 1188: 1179: 1174: 1171: 1170: 1169: 1142: 1141: 1138: 1134: 1133: 1132: 1131: 1130: 1129: 1128: 1127: 1126: 1125: 1124: 1123: 1099: 1098: 1097: 1096: 1095: 1094: 1093: 1092: 1091: 1090: 1080: 1060: 1059: 1058: 1057: 1056: 1055: 1054: 1053: 1032: 1031: 1030: 1029: 1028: 1027: 1026: 1025: 1022:Isolated point 1011: 1010: 1009: 1008: 1007: 1006: 987: 986: 985: 984: 967: 966: 965: 964: 963:come out to 1. 957: 956: 955: 954: 909: 908: 907: 906: 900: 899: 898: 897: 879: 878: 877: 876: 869: 868: 842: 839: 823: 820: 811: 809: 808: 807: 806: 793: 792: 778: 777: 750: 749: 742: 741: 737: 734: 723: 720: 716:www.wbabin.net 711: 708: 707: 706: 634: 631: 606: 605: 587: 586: 556: 555: 554: 553: 537: 536: 535: 534: 533: 532: 511: 510: 494:205.147.225.20 490: 489: 488: 487: 486: 485: 454: 453: 452: 451: 450: 449: 430: 429: 428: 427: 407: 406: 391: 390: 368: 367: 359: 356: 355: 354: 343: 341: 340: 318: 314: 310: 303: 298: 295: 292: 288: 284: 281: 278: 275: 272: 245: 242: 241: 240: 239: 238: 237: 236: 235: 234: 217: 216: 215: 214: 213: 212: 204: 203: 202: 201: 189: 188: 171: 170: 160: 157: 142: 139: 136: 131: 127: 123: 120: 117: 113: 109: 106: 103: 91: 88: 85: 84: 79: 74: 67: 62: 52: 51: 34: 23: 15: 14: 13: 10: 9: 6: 4: 3: 2: 2946: 2937: 2936: 2932: 2928: 2920: 2918: 2917: 2913: 2909: 2908:Mike Schwartz 2905: 2902:was given by 2901: 2897: 2889: 2887: 2885: 2881: 2877: 2876:88.178.225.58 2873: 2864: 2860: 2856: 2852: 2849: 2846: 2843: 2842: 2841: 2838: 2835: 2832: 2825: 2823: 2822: 2818: 2814: 2810: 2805: 2804: 2800: 2796: 2787: 2785: 2784: 2780: 2776: 2772: 2767: 2764: 2762: 2757: 2750: 2748: 2747: 2743: 2739: 2731: 2729: 2728: 2725: 2717: 2713: 2709: 2705: 2701: 2700: 2699: 2698: 2695: 2691: 2687: 2679: 2676: 2672: 2668: 2664: 2660: 2656: 2655: 2654: 2653: 2649: 2645: 2641: 2632: 2630: 2627: 2623: 2619: 2618:85.85.100.144 2615: 2591: 2588: 2585: 2579: 2573: 2570: 2567: 2557: 2552: 2548: 2545: 2539: 2535: 2532: 2527: 2524: 2521: 2517: 2511: 2507: 2503: 2497: 2491: 2484: 2483: 2479: 2478: 2477: 2474: 2470: 2466: 2462: 2452: 2448: 2445: 2441: 2437: 2436: 2435: 2434: 2431: 2427: 2419: 2415: 2412: 2407: 2406: 2403: 2400: 2395: 2394: 2393: 2392: 2389: 2385: 2377: 2375: 2374: 2371: 2365: 2363: 2358: 2357: 2335: 2331: 2327: 2324: 2318: 2312: 2309: 2303: 2299: 2295: 2292: 2279: 2275: 2271: 2265: 2258: 2254: 2250: 2246: 2239: 2233: 2223: 2222: 2221: 2220: 2217: 2214: 2213: 2210: 2207: 2206: 2178: 2172: 2169: 2165: 2158: 2155: 2152: 2144: 2140: 2136: 2131: 2128: 2124: 2117: 2113: 2109: 2105: 2099: 2093: 2090: 2083: 2079: 2075: 2067: 2063: 2059: 2056: 2052: 2048: 2043: 2039: 2035: 2031: 2027: 2021: 2015: 2012: 2005: 2001: 1997: 1994: 1990: 1984: 1980: 1971: 1970: 1969: 1968: 1965: 1962: 1961: 1960: 1954: 1952: 1951: 1948: 1944: 1942: 1935: 1932: 1928: 1927: 1926: 1923: 1922: 1918: 1915: 1912: 1911: 1907: 1901: 1899: 1896: 1895: 1892: 1888: 1881: 1877: 1873: 1869: 1868: 1867: 1866: 1863: 1855: 1852: 1836: 1833: 1827: 1824: 1821: 1817: 1793: 1790: 1787: 1782: 1779: 1772: 1769: 1766: 1762: 1754: 1748: 1745: 1732: 1728: 1724: 1721: 1718: 1715: 1709: 1706: 1703: 1699: 1694: 1688: 1682: 1673: 1670:According to 1669: 1668: 1663: 1659: 1655: 1649: 1648: 1647: 1644: 1640: 1636: 1632: 1623: 1608: 1605: 1602: 1599: 1595: 1588: 1585: 1579: 1563: 1559: 1555: 1552: 1546: 1540: 1532: 1529: 1526: 1522: 1518: 1514: 1503: 1499: 1495: 1491: 1487: 1483: 1482: 1479: 1476: 1472: 1471: 1470: 1467: 1462: 1461: 1457: 1453: 1446: 1442: 1440: 1437: 1433: 1429: 1428:85.85.100.144 1425: 1414: 1412: 1411: 1408: 1404: 1400: 1392: 1390: 1387: 1383: 1379: 1375: 1364: 1362: 1360: 1357: 1352: 1348: 1345: 1339: 1337: 1336: 1332: 1328: 1320: 1317: 1313: 1309: 1305: 1304: 1303: 1301: 1299: 1294: 1290: 1287: 1280: 1275: 1268: 1266: 1263: 1262: 1259: 1234: 1231: 1225: 1222: 1219: 1213: 1210: 1204: 1201: 1195: 1192: 1186: 1178: 1172: 1168: 1165: 1161: 1160: 1159: 1158: 1155: 1151: 1147: 1139: 1136: 1135: 1122: 1119: 1115: 1111: 1110: 1109: 1108: 1107: 1106: 1105: 1104: 1103: 1102: 1101: 1100: 1089: 1086: 1081: 1078: 1074: 1070: 1069: 1068: 1067: 1066: 1065: 1064: 1063: 1062: 1061: 1052: 1049: 1044: 1040: 1039: 1038: 1037: 1036: 1035: 1034: 1033: 1023: 1019: 1018: 1017: 1016: 1015: 1014: 1013: 1012: 1005: 1002: 998: 993: 992: 991: 990: 989: 988: 983: 980: 976: 971: 970: 969: 968: 961: 960: 959: 958: 952: 948: 944: 940: 936: 932: 928: 924: 920: 916: 913: 912: 911: 910: 904: 903: 902: 901: 895: 891: 887: 883: 882: 881: 880: 873: 872: 871: 870: 865: 864: 863: 862: 859: 855: 849: 848: 838: 837: 834: 831: 830: 821: 819: 818: 815: 805: 802: 797: 796: 795: 794: 791: 788: 784: 780: 779: 775: 770: 766: 765: 764: 763: 760: 756: 753: 746: 735: 733: 731: 730: 721: 719: 717: 709: 705: 702: 697: 696: 695: 694: 691: 687: 686: 680: 678: 674: 673: 660: 656: 655:jijin+machina 642: 640: 638: 632: 630: 629: 625: 621: 615: 614: 611: 604: 601: 598: 593: 589: 588: 585: 581: 577: 573: 568: 567: 566: 565: 562: 551: 550: 549: 546: 542: 541: 540: 531: 528: 524: 523: 522: 519: 515: 514: 513: 512: 509: 506: 501: 500: 499: 498: 495: 484: 481: 476: 475: 474: 471: 467: 466: 465: 464: 463: 462: 459: 448: 444: 440: 436: 435: 434: 433: 432: 431: 426: 423: 419: 415: 414:every integer 411: 410: 409: 408: 405: 401: 397: 393: 392: 389: 386: 382: 381: 380: 379: 376: 371: 365: 364: 363: 357: 353: 350: 346: 345: 344: 339: 336: 316: 312: 308: 296: 293: 290: 286: 282: 276: 270: 262: 261: 260: 259: 256: 252: 243: 233: 230: 225: 224: 223: 222: 221: 220: 219: 218: 210: 209: 208: 207: 206: 205: 200: 197: 193: 192: 191: 190: 187: 183: 177: 176: 175: 167: 166: 165: 158: 156: 155: 140: 137: 129: 125: 121: 118: 115: 111: 107: 101: 89: 83: 80: 78: 75: 72: 68: 66: 63: 61: 58: 57: 49: 45: 41: 40: 35: 28: 27: 19: 2924: 2921:A few roots? 2893: 2867: 2855:Jitse Niesen 2839: 2836: 2833: 2829: 2808: 2806: 2791: 2770: 2768: 2765: 2758: 2754: 2735: 2732:Simple zeros 2721: 2704:Jitse Niesen 2702:Checked. -- 2690:Arthur Rubin 2683: 2663:Jitse Niesen 2636: 2609: 2456: 2426:this article 2423: 2383: 2381: 2368: 2359: 2216: 2208: 1963: 1958: 1945: 1940: 1938: 1924: 1919: 1916: 1913: 1908: 1905: 1897: 1884: 1872:Jitse Niesen 1858: 1624: 1533: 1530: 1507: 1465: 1463: 1458:referred by 1454:linked from 1449: 1418: 1402: 1396: 1368: 1353: 1349: 1346: 1343: 1323: 1296: 1285: 1278: 1276: 1272: 1264: 1254: 1176: 1145: 1143: 1042: 974: 950: 946: 942: 938: 934: 930: 926: 922: 918: 914: 893: 889: 885: 875:accordingly. 853: 850: 845: 844: 826: 825: 810: 773: 757: 754: 739: 726: 725: 713: 688: 683: 681: 675: 643: 636: 616: 607: 591: 576:Jitse Niesen 557: 538: 491: 455: 417: 413: 396:Jitse Niesen 372: 369: 361: 342: 249: 247: 185: 172: 162: 93: 70: 43: 37: 2870:—Preceding 2420:A Disproof? 1652:—Preceding 1629:—Preceding 1393:Delisted GA 1378:86.6.13.136 1173:Equivalents 1073:limit point 633:In the News 620:Phoenix1177 610:70.77.11.80 36:This is an 2644:Salix alba 1806:0}" /: --> 1517:88.86.31.1 1314:above. -- 1258:Billymac00 1150:almost all 358:An answer? 335:Rljacobson 2904:Guy Robin 1891:CloudNine 1862:Tenebrous 1672:this page 1443:Proof by 1085:GTBacchus 1043:shouldn't 953:, S5, ... 814:He Who Is 787:GTBacchus 690:Walts0042 677:Walts0042 545:GTBacchus 527:GTBacchus 505:GTBacchus 470:Retinarow 458:GTBacchus 422:GTBacchus 375:Retinarow 82:Archive 5 77:Archive 4 71:Archive 3 65:Archive 2 60:Archive 1 2872:unsigned 2626:contribs 2614:unsigned 2473:contribs 2461:unsigned 2440:Schissel 2430:Gmalivuk 1851:Gmalivuk 1678:0}": --> 1654:unsigned 1643:contribs 1631:unsigned 1513:unsigned 1475:Mrhawley 1424:unsigned 1399:WP:WIAGA 1386:contribs 1374:unsigned 1277:How the 1164:Dmharvey 1146:majority 1114:Dmharvey 1001:Dmharvey 659:Chat Me! 561:pstudier 518:pstudier 2927:Albmont 1943:zeros. 1361:George 743:1. The 480:Tabaqui 349:Ebrevdo 39:archive 2718:Cramer 2694:(talk) 2411:Karl-H 1941:simple 1635:Karl-H 1626:s: --> 1407:Tarret 1118:Calair 1048:Calair 979:Calair 937:, S4, 925:, S2, 917:, S1, 858:Calair 847:strip. 600:(Talk) 255:Hearth 2831:: --> 2370:linas 1791:: --> 1202:<= 975:which 833:linas 801:Jibal 759:Eledu 701:linas 618:typo. 385:Jibal 251:line. 229:linas 196:linas 16:< 2931:talk 2912:talk 2880:talk 2859:talk 2817:talk 2813:Daqu 2799:talk 2795:Daqu 2779:talk 2775:Daqu 2742:talk 2708:talk 2686:edit 2667:talk 2648:talk 2640:link 2622:talk 2469:talk 1902:Misc 1876:talk 1662:talk 1639:talk 1521:talk 1488:and 1432:talk 1382:talk 1331:talk 1327:Daqu 1310:and 1279:hell 637:New 624:talk 580:talk 443:talk 439:Daqu 400:talk 2894:At 2692:| 2533:sin 2399:LDH 1972:1) 1947:LDH 1931:LDH 1405:). 951:L10 894:are 854:not 670:-- 641:]. 597:C S 2933:) 2914:) 2882:) 2861:) 2819:) 2801:) 2781:) 2744:) 2710:) 2669:) 2650:) 2624:• 2589:− 2580:ζ 2571:− 2562:Γ 2546:π 2536:⁡ 2525:− 2518:π 2492:ζ 2471:• 2442:| 2328:π 2288:∞ 2283:∞ 2280:− 2276:∫ 2272:∼ 2266:π 2182:^ 2137:− 2100:− 2080:ψ 2057:− 2049:− 1981:∑ 1889:. 1878:) 1825:− 1800:0} 1788:σ 1749:− 1738:∞ 1729:∫ 1722:− 1707:− 1683:ζ 1664:) 1645:) 1641:• 1606:− 1600:− 1586:− 1569:∞ 1560:∫ 1541:ζ 1523:) 1434:) 1384:• 1333:) 1316:EJ 1295:: 1291:- 1220:∗ 1046:-- 949:, 947:L9 945:, 943:L8 941:, 939:L7 935:L6 933:, 931:L5 929:, 927:L4 923:L3 921:, 919:L2 915:L1 890:L2 888:, 886:L1 772:1 657:| 626:) 582:) 445:) 402:) 302:∞ 287:∑ 271:ζ 102:ζ 2929:( 2910:( 2878:( 2857:( 2815:( 2797:( 2777:( 2740:( 2706:( 2665:( 2646:( 2620:( 2595:) 2592:s 2586:1 2583:( 2577:) 2574:s 2568:1 2565:( 2558:) 2553:2 2549:s 2540:( 2528:1 2522:s 2512:s 2508:2 2504:= 2501:) 2498:s 2495:( 2467:( 2339:) 2336:4 2332:/ 2325:+ 2322:) 2319:x 2316:( 2313:V 2310:u 2307:( 2304:i 2300:e 2296:x 2293:d 2259:2 2255:/ 2251:1 2247:u 2243:) 2240:u 2237:( 2234:Z 2190:) 2179:H 2173:u 2170:i 2166:e 2162:( 2159:r 2156:T 2153:= 2145:u 2141:e 2132:u 2129:3 2125:e 2118:2 2114:/ 2110:u 2106:e 2094:u 2091:d 2084:0 2076:d 2068:2 2064:/ 2060:u 2053:e 2044:2 2040:/ 2036:u 2032:e 2028:= 2025:) 2022:u 2019:( 2016:Z 2013:= 2006:n 2002:E 1998:u 1995:i 1991:e 1985:n 1874:( 1837:1 1834:+ 1828:1 1822:s 1818:1 1794:0 1783:x 1780:d 1773:1 1770:+ 1767:s 1763:x 1758:] 1755:x 1752:[ 1746:x 1733:1 1725:s 1719:1 1716:+ 1710:1 1704:s 1700:1 1695:= 1692:) 1689:s 1686:( 1660:( 1637:( 1609:1 1603:s 1596:x 1592:) 1589:x 1583:] 1580:x 1577:[ 1574:( 1564:0 1556:s 1553:= 1550:) 1547:s 1544:( 1527:. 1519:( 1438:. 1430:( 1380:( 1329:( 1300:! 1284:— 1238:) 1235:n 1232:H 1229:( 1226:n 1223:l 1217:) 1214:n 1211:H 1208:( 1205:e 1199:) 1196:Q 1193:A 1190:( 1187:e 1083:- 668:× 665:× 663:× 651:× 649:× 646:× 622:( 578:( 503:- 441:( 398:( 317:s 313:n 309:1 297:1 294:= 291:n 283:= 280:) 277:s 274:( 141:0 138:= 135:) 130:n 126:E 122:i 119:+ 116:2 112:/ 108:1 105:( 50:.

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index