This Thursday I was at the University of Sydney, Australia, giving a public lecture on a favourite topic of mine, “Structure and randomness in the prime numbers“. My slides here are a merge between my slides for a Royal Society meeting and the slides I gave for the UCLA Science Colloquium; now that I figured out to use Powerpoint a little bit better, I was able to make the latter a bit more colourful (and the former less abridged).
Recent Comments
The Polymath Blog
- Proposals (Tim Gowers): Polynomial DHJ, and Littlewood’s problem
- Proposal (Tim Gowers) The Origin of Life
- (Research thread V) Determinstic way to find primes
- Nature article on Polymath
- (Research Thread IV) Determinstic way to find primes
- (Research Thread III) Determinstic way to find primes
- (Research thread II) Deterministic way to find primes
- Polymath on other blogs
- Deterministic way to find primes: discussion thread
- Selecting another polymath project
Mathematics in Australia
- Positions at Australian National University 4 November, 2009
- Mathematics skills out for the count 26 October, 2009
- Federal government awards $2 million for Improving Mathematics Education in Schools project 13 October, 2009
- Message from AMSI regarding proposed cuts at Victoria University 9 October, 2009
- Jurassic Quark 8 October, 2009
- Nobel Prize Laureates 7 October, 2009
- Access to mathematics is vital for equity 29 September, 2009
- Aussie mathematician blogs 26 September, 2009
- ARC releases consultation paper on its peer review process 18 September, 2009
- Professor of Pure Mathematics position at University of Adelaide 17 September, 2009
44 comments
Comments feed for this article
7 February, 2008 at 10:03 am
Anon
Nice slides!
I would recommend to convert them (directly) into
PDF (unless you have significant animation in them).
7 February, 2008 at 10:39 pm
Derek Buchanan
The date on the first page is wrong. It’s the 7th, not the 8th.
Anyway, Terry, I was the one who asked you later about polynomial progressions of primes. I’m curious as to why you did not mention this in your talk.
18 February, 2008 at 7:51 am
mitchan88
Nice slides!
I like your site very much, but unfortunately I’m only at the first year of University, so I can understand only 0,5% of your articles :p
27 February, 2008 at 9:26 am
dsilvestre
Dear Dr Tao,
I watched this video and liked it very much.
Hey, if I want to find a drawing on the gaussian primes constelation, say, this rabbit
do I have a chance? (my mom drew it)
what are the chances of finding it using 1-year of a pentium4 time?
3 March, 2008 at 2:42 pm
alexandru
yes, pdf would be great… especially for linux users…
2 July, 2008 at 11:54 am
Ian Agol
Anyone understand Xian-Jin Li’s paper claiming to prove the Riemann hypothesis?
http://arxiv.org/abs/0807.0090
Look’s like he’s completing a program of Bombieri and Connes.
He seems to be an expert on RH, and has done some nice work on it before.
2 July, 2008 at 2:11 pm
I Can Has Riemann Hypothesis? « The Unapologetic Mathematician
[...] Can Has Riemann Hypothesis? Everybody is talking about [...]
2 July, 2008 at 6:28 pm
Terence Tao
It unfortunately seems that the decomposition claimed in equation (6.9) on page 20 of that paper is, in fact, impossible; it would endow the function h (which is holding the arithmetical information about the primes) with an extremely strong dilation symmetry which it does not actually obey. It seems that the author was relying on this symmetry to make the adelic Fourier transform far more powerful than it really ought to be for this problem.
2 July, 2008 at 7:09 pm
Not Even Wrong » Blog Archive » Proof of the Riemann Hypothesis?
[...] It looks like a problem with the proof has been found. Terry Tao comments on his blog It unfortunately seems that the decomposition claimed in equation (6.9) on page 20 of [...]
2 July, 2008 at 7:47 pm
Another attempted proof of the Riemann hypothesis « Muse Free
[...] I’d think that the chances of this proof being correct are extremely low; in fact Terry Tao claims to have already found a [...]
3 July, 2008 at 3:41 am
Gergely Harcos
I also have some (perhaps milder) troubles with the proof. It seems to me as if Li had treated the Dirac delta on L^2(A) as a function. For example, the first 5 lines of page 28 make little sense to me. Am I missing something here?
3 July, 2008 at 10:04 am
Ars Mathematica » Blog Archive » Li’s Preprint
[...] proof the Riemann Hypothesis. The optics of it looked good (Li is clearly not a crank), but Terry Tao has identified an apparent [...]
3 July, 2008 at 1:58 pm
Terence Tao encuentra un fallo en la demostración de Li de la Hipótesis de Riemann [ENG]
[...] Terence Tao encuentra un fallo en la demostración de Li de la Hipótesis de Riemann [ENG]terrytao.wordpress.com/2008/02/07/structure-and-randomness-i… por emulenews hace pocos segundos [...]
3 July, 2008 at 2:12 pm
Otro valiente más (o una nueva “demostración” de la hipótesis de Riemann) « Francis (th)E mule Science’s News
[...] Fields en el ICM de Madrid de 2006, uno de los matemáticos más geniales vivos en la actualidad) afirma en su blog que ha encontrado ciertos “problemas” con una descomposición presentada en la página [...]
3 July, 2008 at 3:13 pm
Request: Li’s preprint, or “on not being a crackpot” « Secret Blogging Seminar
[...] of us seemed inclined to go through the paper and look for mistakes. Luckily, Terry Tao did and thinks he has found a mistake (which the author may claim to have fixed…things are starting to get a little confusing). [...]
4 July, 2008 at 5:10 am
Richard Elwes - Riemann Hypothesis
[...] Terry Tao and Alain Connes both say “no”. Ok, Li’s apparently “updated” the [...]
4 July, 2008 at 5:15 am
Lior Silberman
The function
defined on page 20 does have a strong dilation symmetry: it is invariant by multiplication by ideles of norm one (since it is merely a function of the norm of 
). In particular, it is invariant under multiplication by elements of 
. I’m probably missing something here.
Probably the subtlety is in passing from integration over the nice space
of idele classes to the singular space 
. The topologies on the spaces of adeles and ideles are quite different.
There is a formal error in Theorem 3.1 which doesn’t affect the paper: the distribution discussed is not unique. A distribution supported at a point is a sum of derivatives of the delta distribution. Clearly there exist many such with a given special value of the Fourier transform.
There is also something odd about this paper: nowhere is it pointed out what is the new contribution of the paper. Specifically, what is the new insight about number theory?
4 July, 2008 at 6:09 am
Emmanuel Kowalski
A remark concerning Lior’s remark: the function h(u) in the current (v4) version of the paper is _not_ the same as the one that was defined when T. Tao pointed out a problem with it. This earlier one (still visible on arXiv, v1) was defined in different ways depending on whether the idele had at most one or more than one non-unit component, and was therefore not invariant under multiplication by
.
(It is another problem with looking at such a paper if corrections as drastic as that are made without any indication of when and why).
4 July, 2008 at 8:15 am
Terence Tao
Dear Lior,
Emmanuel is correct. The old definition of h was in fact problematic for a large number of reasons (the author was routinely integrating h on the idele class group C, which is only well-defined if h was
-invariant). Changing the definition does indeed fix the problem I pointed out (and a number of other issues too). But Connes has pointed out a much more serious issue, in the proof of the trace formula in Theorem 7.3 (which is the heart of the matter, and is what should be focused on in any future revision): the author is trying to use adelic integration to control a function (namely, h) supported on the ideles, which cannot work as the ideles have measure zero in the adeles. (The first concrete error here arises in the equation after (7.13): the author has made a change of variables 
on the idele class group C that only makes sense when u is an idele, but u is being integrated over the adeles instead. All subsequent manipulations involving the adelic Fourier transform Hh of h are also highly suspect, since h is zero almost everywhere on the adeles.)
More generally, there is a philosophical objection as to why a purely multiplicative adelic approach such as this one cannot work. The argument only uses the multiplicative structure of
, but not the additive structure of k. (For instance, the fact that k is a cocompact discrete additive subgroup of A is not used.) Because of this, the arguments would still hold if we simply deleted a finite number of finite places v from the adeles (and from 
). If the arguments worked, this would mean that the Weil-Bombieri positivity criterion (Theorem 3.2 in the paper) would continue to hold even after deleting an arbitrary number of places. But I am pretty sure one can cook up a function g which (assuming RH) fails this massively stronger positivity property (basically, one needs to take g to be a well chosen slowly varying function with broad support, so that the Mellin transforms at Riemann zeroes, as well as the pole at 1 and the place at infinity, are negligible but which gives a bad contribution to a single large prime (and many good contributions to other primes which we delete).)
4 July, 2008 at 8:25 am
Emmanuel Kowalski
That’s an interesting point indeed, if one considers that the RH doesn’t work over function fields once we take out a point of a (smooth projective) curve — there arise zeros of the zeta function which are not on the critical line.
4 July, 2008 at 10:54 am
Update on Riemann hypothesis « Muse Free
[...] Tao explains here why an approach such as Li’s, which basically only uses the multiplicative structure of the [...]
4 July, 2008 at 6:49 pm
Posible demostración de la hipótesis de Riemann « Edumate Perú
[...] debate lo plantea el matemático Terence Tao de la Universidad de los California – Los Ángeles. En su blog plantea que ciertos pasos seguidos por Xian-Jin Li no están del todo bien. It unfortunately seems that the [...]
5 July, 2008 at 5:56 am
Moshe Klein
Dear Terence Tao, 5.7.08
Maybe You and A.Connes can work together to close the gap which you discover in Li new paper on RH.
B.Riemann claim in 1859 that the zeros of the Zeta function lies all on the critical line ( This is one part of the 8 problem of Hilbert) Do you think that an interaction between a line and points as individual
atoms can create a new mathematical framework which can be consider as a
solution to the 6 th’ problem of Hilbert about the connection between mathematics and physics ?
Best Wishes
Moshe Klein
5 July, 2008 at 9:01 am
Desmontada la demostración de Li
[...] en la UCLA y ganador de la medalla Fields (considerada el premio nobel de las matemáticas), ha encontrado un fallo en su demostración. El error está en una fórmula donde se atribuye a los números primos unas [...]
5 July, 2008 at 7:16 pm
Abhishek
Xian-Jin Li has now withdrawn his paper from the arXiv.
6 July, 2008 at 5:28 pm
Chip Neville
Terence,
I have a question about your comment:
“Because of this, the arguments would still hold if we simply deleted a finite number of finite places v from the adeles (and from k^*). … (basically, one needs to take g to be a well chosen slowly varying function with broad support, so that the Mellin transforms at Riemann zeroes, as well as the pole at 1 and the place at infinity, are negligible but which gives a bad contribution to a single large prime (and many good contributions to other primes which we delete).)”
Does this mean that you would be considering the “reduced” (for lack of a better name) zeta function \prod 1/(1-1/p^{-s}), where the product is taken over the set of primes not in a finite subset S? If so, this “reduced” zeta function has the same zeroes as the standard Riemann zeta function, since the finite product \prod_S 1/(1-1/p^{-s}) is an entire function with no zeroes in the complex plane. Thus the classical situation in the complex plane seems to be very different in this regard from the situation with function fields over smooth projective curves alluded to by Emmanuel above.
Does anyone have an example of an infinite set S and corresponding reduced zeta function with zeroes in the half plane Re z > 1/2? A set S of primes p so that \sum_S 1/p^{1/2} converges will not do, since \prod_S 1/(1-1/p^{-s}) is holomorphic in the half plane Re z > 1/2 with no zeroes there. Perhaps a set S of primes P thick enough so that \sum_S 1/p^{1/2} diverges, but thin enough so that \sum_S 1/p converges, might do. This seems to me to be a delicate and difficult matter.
I hope these questions do not sound too foolish.
6 July, 2008 at 7:44 pm
Terence Tao
Dear Chip,
Actually, the product
has a number of poles on the line 
, when s is a multiple of 
.
Li’s approach to the RH was not to tackle it directly, but instead to establish the Weil-Bombieri positivity condition which is known to be equivalent to RH. However, the proof of that equivalence implicitly uses the functional equation for the zeta function (via the explicit formula). If one starts deleting places (i.e. primes) from the problem, the RH stays intact (at least on the half-plane
), but the positivity condition does not, because the functional equation has been distorted.
The functional equation, incidentally, is perhaps the one non-trivial way we do know how to exploit the additive structure of k inside the adeles, indeed I believe this equation can be obtained from the Poisson summation formula for the adeles relative to k. But it seems that the functional equation alone is not enough to yield the RH; some other way of exploiting additive structure is also needed, but I have no idea what it should be.
[Revised, July 7:] Looking back at Li’s paper, I see now that Poisson summation was indeed used quite a few times, and in actually a rather essential way, so my previous philosophical objection does not actually apply here. My revised opinion is now that, beyond the issues with the trace formula that caused the paper to be withdrawn, there is another fundamental problem with the paper, which is that the author is in fact implicitly assuming the Riemann hypothesis in order to justify some facts about the operator E (which one can think of as a sort of Mellin transform multiplier with symbol equal to the zeta function, related to the operator
on 
). More precisely, on page 18, the author establishes that 
and asserts that this implies that 
, but this requires certain invertibility properties of E which fail if there is a zero off of the critical line. (A related problem is that the decomposition 
used immediately afterwards is not justified, because 
is merely dense in 
rather than equal to it.)
6 July, 2008 at 10:30 pm
Riemann hypothesis proved? - Bad Astronomy and Universe Today Forum
[...] make the adelic Fourier transform far more powerful than it really ought to be for this problem. Source. __________________ MacTalk – The Australian Apple Community – iPod, iPhone and [...]
7 July, 2008 at 5:14 am
Felipe Voloch
Chip, look up Beurling primes. It will give a sort of an answer to your question.
7 July, 2008 at 9:59 am
javier
Dear Terence,
I am not sure I understand your “philosophical” complain on using only the multiplicative structure and not the additive one. This is essentially the philosophy while working over the (so over-hyped lately) field with one element, which apparently comes into the game in the description of the Connes-Bost system on the latest Connes-Consani-Marcolli paper (Fun with F_un).
From an algebraic point of view, you can often recover the additive structure of a ring from the multiplicative one provided that you fix the zero. There is an explanation of this fact (using the language of monads) in the (also famous lately) work by Nikolai Durov “A new approach to Arakelov geometry (Section 4.8, on additivity on algebraic monads).
By the way, I wanted to tell you that I think you are doing an impressive work with this blog and that I really enjoy learning from it, even if this is the very first time I’ve got something sensible to say :-)
7 July, 2008 at 11:01 am
Terence Tao
Dear Javier,
I must confess I do not understand the field with one element much at all (beyond the formal device of setting q to 1 in any formula derived using
and seeing what one gets), and don’t have anything intelligent to say on that topic. Regarding my philosophical objection, the point was that if one deleted some places from the adele ring A and the multiplicative group 
(e.g. if k was the rationals, one could delete the place 2 by replacing 
with the group of non-zero rationals with odd numerator and denominator) then one would still get a perfectly good “adele” ring in place of A, and a perfectly good multiplicative group in place of 
(which would be the invertible elements in the ring of rationals with odd denominator), but somehow the arithmetic aspects of the adeles have been distorted in the process (in particular, Poisson summation and the functional equation get affected). The Riemann hypothesis doesn’t seem to extend to this general setting, so that suggests that if one wants to use adeles to prove RH, one has to somehow exploit the fact that one has all places present, and not just a subset of such places. Now, Poisson summation does exploit this very fact, and so technically this means that my objection does not apply to Li’s paper, but I feel that Poisson summation is not sufficient by itself for this task (just as the functional equation is insufficient to resolve RH), and some further exploitation of additive (or field-theoretic) structure of k should be needed. I don’t have a precise formalisation of this feeling, though.
7 July, 2008 at 1:22 pm
Gergely Harcos
Dear Terry,
you are absolutely right that Poisson summation over k inside A is the (now) standard way to obtain the functional equation for Hecke L-functions. This proof is due to Tate (his thesis from 1950), you can also find it in Weil’s Basic Number Theory, Chapter 7, Section 5.
9 July, 2008 at 1:19 am
j.
Why don’t you use LaTeX for your slides?
14 July, 2008 at 1:02 am
Babak
Hi Terrance,
A few months ago I stumbled upon an interesting differential equation while using probability heuristics to explore the distribution of primes. It’s probably nothing, but on the off-chance that it might mean something to a better trained mind, I decided to blog about it: http://babaksjournal.blogspot.com/2008/07/differential-equation-estimating.html
-Babak
15 July, 2008 at 7:57 am
michele
I think that the paper of Prof. Xian-Jin Li will be very useful for a future and definitive proof of the Riemann hypothesis. Furthermore, many mathematics contents of this paper can be applied for further progress in varios sectors of theoretical physics (p-adic and adelic strings, zeta strings).
17 July, 2008 at 2:29 am
Yonghui Wang
“functional equation is insufficient to resolve RH”
Yes, For example, The Hurwitz zeta-function and the Eisenstein series (See
Lagarias paper or my paper on Acta Math Hung.~mine is very easy, just for
an test) do not satisfy RH, though they have functional equation.
The point is that, they do not have Euler product! ~a multiplicative
structure!
By the way, the Possion Summation on Adele do use the fact that ” k is a
cocompact discrete additive subgroup of A”, which takes the role as the
Fourier analysis on R/Z. This certifies that “Functional equation” (deduced
from Poisson Summation) represents the additive structure.
So, the philosophy of “multiplicative” and “additive” is really critical for
analytic number theorist.
Usually, the analytic number theorist believe that a general L-function
which satisfies both functional equation and Euler Products will posses RH.
5 August, 2008 at 1:27 am
Qiaochu Yuan
Professor Tao, I’m not quite sure where to put this comment, but I was wondering whether your results with Ziegler on polynomial progressions in the primes implied or could be used to deduce effective bounds that would help explain patterns like the Ulam spiral. In other words, can it be deduced that certain quadratic patterns (such as the well-known n^2 + n + 41) appear more often in the primes than “expected” (for an appropriate definition of “expected”) from your results?
Since that particular example is related to the Heegner numbers, I would be pleasantly surprised if the answer was affirmative, although from what I’ve read the techniques you apply do not actually use much information about the primes themselves, so perhaps it’s not philosophically likely.
5 August, 2008 at 9:41 am
Terence Tao
Dear Qiaochu,
My result with Tamar is only able to deal with polynomial expressions involving two or more variables, e.g. n + P(r) where n, r are integers and P is a polynomial. It is still an open problem whether there is even a single non-linear polynomial P such that P(n) is prime infinitely often (though it is widely believed that any such polynomial which is coprime to any given modulus infinitely often, such as n^2+n+41, will in fact capture infinitely many primes, cf. Schinzel’s hypothesis H).
In our paper, we do look a little bit at the “local factors” of a polynomial such as P(n) or P(n,r) with respect to a given modulus q, and in particular how often the polynomial is coprime to q. For certain quadratics such as n^2+n+41, this factor is slightly higher for many small q than for generic quadratics, which may at least partially explain the Ulam spiral phenomenon. But we did not need very precise bounds on these local factors, and we ended up just using extremely crude bounds from baby algebraic geometry. (At one point we were worried that we would have to use some version of the BSD conjecture, though this turned out (fortunately) to be based on a misconception as to where the pole of a certain zeta function was.)
6 December, 2008 at 2:40 pm
The uniform uncertainty principle and compressed sensing « What’s new
[...] and randomness in the prime numbers. This lecture is largely equivalent to the one posted here.] Possibly related posts: (automatically generated)Law School Prof: “Never, Ever Talk to a [...]
11 December, 2008 at 2:12 am
Quotes on primes « Asymptotics
[...] makes a mathematician go beyond theorems and proofs to praise them. A couple of quotes I found on Terrence Tao’s blog . The first one is based on the reductio ad absurdum proof for the existence of infinitely many [...]
20 January, 2009 at 6:16 pm
Más sobre la posible demostración de la hipótesis de Riemann
[...] les dejo un comentarios de vengoroso donde explica las fallas: Traduzco un comentario de Terry Tao en su propio blog: Desafortunadamente parece que la descomposición propuesta en la ecuación (6.9) de la página 20 [...]
6 September, 2009 at 7:27 pm
Two more Clay-Mahler lectures « What’s new
[...] and randomness in the prime numbers“. This public lecture is slightly updated from a previous talk of the same name given last year, but is largely the same [...]
30 September, 2009 at 1:45 am
Ian Robinson
Dear Professor Tao
Good morning, my name is Ian Robinson living on the east coast of Australia. I am contacting you with regards to the distribution of Prime Numbers. There seems to be two accepted facts about their distribution, one is that they do seem to grow like weeds among the natural numbers, and two they do have stunning regularity with laws governing their behaviour with military precision.
I am writing with a humble air of confidence because yesterday I re-discovered the true distribution of Primes and their supporting Composites.
The Prime Numbers are responsible for the physical structure of in this instance the ‘Whirlpool Galaxy.’ In a nutshell the Prime Numbers are strategically placed on the spiralling arms of the galaxy, they can be seen as naturally plotted prominent star clusters on the spiralling arms that give physical structure to the trunk and branches. I achieved this discovery by measuring the distance of each prime number from a centre point then plotted the distance down onto paper whilst moving in a clockwise direction,I then joined up the dots to complete the spiralling picture of Prime Numbers. I repeated the process with the composite numbers and joined up the dots that spiralled outwards with stunning regularity resembling a spiders web, yet both sets of numbers are working in complete unison.
The image of the physical makeup of the Whirlpool Galaxy looks identical to my drawing, I have deliberately withheld an important piece of the jigsaw until a later date.
The truth is I require someone with a bit of mathematical clout to give these findings the validity and respect they deserve, no one else is aware of my findings.
A timely response will be much appreciated Professor Tao and believe me this discovery is for real and not a fabricated hoax.
Thank you
Regards
Ian Robinson
8 November, 2009 at 4:52 am
Struktur dan keteracakan bilangan prima « Proof { }
[...] Struktur dan keteracakan bilangan prima 2009 February 19 tags: bilangan, matematika, Math, prima, struktur by Aria Turns Structure and randomness in the prime numbers [...]