12 ms·

A breakthrough towards the Riemann hypothesis

540 points by pera 2y ago

ganzuul 2y ago
This improves modeling of long-tail distributions? How far off am I here?
- gus_massa 2y ago
  If the complex number is x+iy: They had a good bound of the long-tail distribution when x>=3/5=0.6. Now someone extended that result to x>=13/25=0.52. (The long term objective is to prove a stronger version for x>1/2=.5.)
  - deleted 2y ago
    [deleted]
  - konstantinua00 2y ago
    it's different affected distribution is always x>3/4, both before and after what's measured is upper bound on number of zeroes <y, relative to y it was <y^(3/5), now it's <y^(13/25) it says nothing about absence of zeroes, but the density result already affects prime distributions
    - gus_massa 2y ago
      My bad. Thanks for the correction.
- lupire 2y ago
  Can you be more specific? This is a result about a specific approximately known distribution.
  - ganzuul 2y ago
    Excuse me I meant the related heavy-tailed distributions, which show up in dragon king theory. I'm wondering if this result indicates we have a new method to eke out important signals from noise that otherwise get smoothed out.
jl6 2y ago
Without intending to take anything away at all from the work of the researchers, I think “breakthrough” might just be overselling it. They have improved a bound but I can’t tell that their method will open up a plausible path to a full proof. It feels like we’ve managed to increase the top speed of a car from 200mph to 227mph, but are no closer to understanding how the engine works.
- magnio 2y ago
  Change that to CPU performance in 2023 and you got a breakthrough as well.
- math_dandy 2y ago
  It may take many breakthroughs to yield a proof of the Riemann Hypothesis. Progress on a notoriously hard problem through introduction of innovative new ideas can certainly qualify as a breakthrough, imo. Naturally, the importance of the breakthrough will be judged by what can be built upon it.
  - fishtockos 2y ago
    This is clever application of existing ideas, not the sort of new idea which is needed. No one can really imagine what sort of machinery will need to be built to prove the RH. A proof is completely outside the realm of speculation at this point.
- btilly 2y ago
  When you've been stuck for 80 years, any improvement at all seems like a big deal. But more importantly, a new idea like this often points the way to variations that may also create improvement. Therefore as other researchers pile on, we may see this bound come down further. So yes, it is like a wall in a maze was breached, and there is the opportunity to go further. And it is fair to call that a breakthrough. Of course you're still charging into a maze, and further progress is far from guaranteed.
- gus_massa 2y ago
  The author of the post is Terence Tao that is the best live mathematician. If he says it's a "breakthrough", it's a breakthrough. (Anyway, I agree that if you are not in this research area, it's still an internal result that you can safetly ignore.) Also, in the last years he was leading a few project to crowsource improvements after breakthroughs in other conjetures. I'm too far from this area to be sure, but if he sees there is a chance to improve this, I expect him to launch a new one. From the article: > but they do a number of clever and unexpected maneuvers If he post something like that about me, I'd frame it an hang it in the wall.
pkilgore 2y ago
Found this[1] a useful primer on potential significance from a 2018 proposed proof. [1] https://www.sciencenews.org/article/why-we-care-riemann-hypothesis-math-prime-numbers https://www.sciencenews.org/article/why-we-care-riemann-hypo...
- nomilk 2y ago
  Also curious about the potential significance of a proof. The article is vague: > (primes) are important for securing encrypted transmissions sent over the internet. And importantly, a multitude of mathematical papers take the Riemann hypothesis as a given. If this foundational assumption were proved correct, “many results that are believed to be true will be known to be true,” says mathematician Ken Ono of Emory University in Atlanta. “It’s a kind of mathematical oracle.” Are there some obvious, known applications where a RH proof would have immediate practical effects? (beyond satisfaction and 'slightly better encryption').
  - dgacmu 2y ago
    If we knew that the extended Riemann hypothesis was true, we could use the Miller test for deterministic primality testing in log(n)^4; the AKS test, which doesn't depend on RH, is lg(n)^6. Do we care? Not for most applications -- doing a bunch of randomized Miller-Rabin tests is fine for most practical purposes. But it would be really nice to have a faster deterministic algorithm around. AKS isn't practical for anything; miller... Miiiiiggghtt be.
    - olddustytrail 2y ago
      Then why not just assume it's true and if something breaks, hey, you might just have disproved it?
      - tekchip 2y ago
        [flagged]
      - aj7 2y ago
greekanalyst 2y ago
Always amazed by Terence Tao's clarity of thought. Not only is he one of the greatest mathematical minds alive (if not the greatest), he is also one of the most eloquent writers on mathematics. Nothing more beautiful than seeing great science being married with great writing. Even if you don't understand the specifics, you can always get the big picture. Thank you, Terence!
- alexander2002 2y ago
  Does he have any beginner friendly books on math I saw some clips of his masterclass but I am skeptical of masterclass in general so I am not sure about that
  - dan-robertson 2y ago
    He has a blog which has some more elementary posts and some less. But I’m not sure if that would be sufficiently ‘beginner friendly’ for your needs. https://terrytao.wordpress.com/ https://terrytao.wordpress.com/
  - ted_dunning 2y ago
    His masterclass is substantially better than the average masterclass, particularly for somebody who isn't already an amateur or better mathematician.
  - empeyot 2y ago
    His book Analysis I, and potentially also his linear algebra lecture come to mind.
idlewords 2y ago
Trying to imagine what it must feel like to have Terence Tao summarize your argument while mentioning that he'd tried something similar but failed. "The arguments are largely Fourier analytic in nature. The first few steps are standard, and many analytic number theorists, including myself, who have attempted to break the Ingham bound, will recognize them; but they do a number of clever and unexpected maneuvers."
- mi_lk 2y ago
  I mean, two authors of the paper are pretty well established in the field already [0]: https://en.wikipedia.org/wiki/Larry_Guth https://en.wikipedia.org/wiki/Larry_Guth [1]: https://en.wikipedia.org/wiki/James_Maynard_(mathematician) https://en.wikipedia.org/wiki/James_Maynard_(mathematician)
  - chx 2y ago
    Yes but it's Terence Tao. I mean, the set of living mathematicians is not well ordered on greatness but if it were, Terence Tao would be fairly close to the upper limit.
    - Ar-Curunir 2y ago
      Maynard is a Fields Medallist, so is also one of the strongest mathematicians around.
    - alkyon 2y ago
      Maynard won a Fields medal, Tao is of course in the elite, but so is JM.
- deleted 2y ago
  [deleted]
- nybsjytm 2y ago
  It's perfectly common for a mathematician to successfully use a technique where another top mathematician tried and failed.
mr-roboto 2y ago
Good timing for me. I'm in the middle of listening to "The Humans" by Matt Haig. The story opens after someone proves the Riemann hypothesis.
vsnf 2y ago
What are the practical implications of a conclusion of the hypothesis, in either direction?
- throw_pm23 2y ago
  This is pure math, so not much, at least directly or immediately. But I find it amusing how this argument always comes up and how it goes back millenia. A student of Plato (428 B.C. -- 348 B.C.) once asked the great master, "What practical uses do these theorems serve? What is to be gained from them?" Plato's answer was immediate and peremptory. He turned to one of his slaves and said, "Give this young man an obol [a small Greek coin] so that he may feel that he has gained something from my teachings. Then expel him."
  - vsnf 2y ago
    I didn’t mean to imply that it needed to be useful. I just hear about the hypothesis a lot and I wonder what the immediate knock-on effects would be. Does it unlock other theoretical work, are there other proofs that work or don’t work depending on it, etc. And if it has an effect on something real or tangible that’s even better.
    - moi2388 2y ago
      In principle a proof of the Riemann Hypothesis could give you information about the distribution of primes and could possible make it easier to test for primality, in the worst case breaking modern encryption. But that’s a lot of what ifs away.
      - Ar-Curunir 2y ago
        There is no known result that says RH being true breaks modern encryption; if there was, the cryptanalysts would assume RH and try to break it anyway.
        IsTom 2y ago
wing-_-nuts 2y ago
Can I get a ELI!M[Not a mathematician]?
- nhatcher 2y ago
  One of the most important open problems in mathematics is called the Riemann hypothesis. It states that the solutions of a certain equation `zeta(z)=0` are all of a particular type. Almost every living mathematician has tried to solve it at some point in their lives. The implications of the hypothesis are deep for the theory of numbers, for instance for the distribution of prime numbers. In a recent paper some mathematicians claim they have put some stronger bounds on where those solutions can be. In this link Terrence Tao, one of the most acclaimed mathematicians alive speaks very highly about the paper. IMHO, this is probably not of huge interest to not mathematicians just yet. It is an extremely technical result. And pending further review it might very well be wrong or incomplete. There are lots of places you can read about the Riemann Hypothesis, its implications and its attempts to solve it.
  - chii 2y ago
    As for why the Riemann Hypothesis itself is interesting, it is because this zeta function seems to have information about the primes inside it. The eye opener for me is when a video about it connected the idea of the zeta function to how you might create a square wave by adding successive sine waves (of higher and higher frequency). Imagine a function (call it `prime_count`), which gives you the number of prime numbers below it's argument. If you tried to plot this function on a graph, it will look like a series of jaggard steps - a jump when you reach a new prime number. It turns out, if you rewrite the zeta function and decompose it (into it's infinitely many zeros - akin to doing a "taylor series" expansion for other functions), you will find that the more zeros you use, the closer you will get to a jaggard graph. This video is really good at giving a visual explanation : https://youtu.be/e4kOh7qlsM4?t=594 https://youtu.be/e4kOh7qlsM4?t=594
    - seanhunter 2y ago
      > The eye opener for me is when a video about it connected the idea of the zeta function to how you might create a square wave by adding successive sine waves (of higher and higher frequency). The process of taking an arbitrary waveform and approximating it by adding together a series of sine and/or cosine waves with different amplitudes etc is called a Fourier transform. That is what Terrence Tao is talking about when he mentions trying this approach using Fourier analysis himself. [1] Fourier transforms are used all over the place, eg the discrete cosine transform (DCT) is part of JPEG, MP3 etc. [1] https://en.wikipedia.org/wiki/Fourier_analysis https://en.wikipedia.org/wiki/Fourier_analysis
pheatherlite 2y ago
[flagged]
- ipnon 2y ago
  I like how Tao can just write a few quick candid thoughts and post them online for everyone.
- jcla1 2y ago
  I think giving any more details than his three short posts would delve too far into the specific arguments of the paper to be valuable for a wide enough audience, subtracting those that are going to be looking at the pre-print anyhow.
- rrauenza 2y ago
  I have discovered a truly marvelous demonstration of this proposition that .... will not fit in 280 characters. -- Fermat
- deleted 2y ago
  [deleted]
cvoss 2y ago
What are your opinions of all the theorems that rely on RH as an excluded middle? Constructivists reject exmid, saying instead that a proof of "A or B" requires you to have in hand a proof of A or a proof of B. And nobody yet has a proof of RH nor a proof of ~RH. This is important in so-called incomplete logical systems, where some theorems are neither provable nor disprovable, and, therefore, exmid is an inadmissible axiom.
- nerdponx 2y ago
  Aren't those sort of different things? I thought the whole point of provability was that it was distinct from truth.
  - floxy 2y ago
    How do you know something is true if you don't have a proof? It all depends on you views on the philosophy of mathematics. Are there "true" statements that don't have proofs? Some say yes, there are platonic ideas that are true, even if they aren't provable. Others say, "what does it even mean to say something is true, if there is no proof. What you really have is a conjecture."
    - lisper 2y ago
      There is another possibility: it could be an arbitrary choice, as in the case of the parallel postulate, the axiom of choice, and non-standard models of the Peano axioms.
      - btilly 2y ago
        According to logicians, it always is an arbitrary choice. But we have a second-order notion of what "finite integer" should mean. And within that notion the idea might be false. Here's how that plays out. Suppose that the RH cannot be proved or disproved from ZF. (It turns out that choice cannot matter for all theorems in number theory, so ZF is enough.) That means that there is a model of ZF in which RH is true. Every model of ZF contains a finite calculation for any non-trivial zero of the Riemann function. (The word "finite" is doing some heavy lifting here - it is a second order concept.) That calculation must work out the same in all models. Therefore every finite nontrivial zero has complex part 0.5. Therefore RH is actually true of the standard model of the integers. Therefore every model of ZF where RH is false, is non-standard. The truth of RH is therefore independent of ZF. But it's true in our second order notion of what the integers are.
greenthrow 2y ago
This comments section is very oddly filled with people who don't actually understand the subject matter but want to sound smart, and then accomplish the opposite. Let go of your insecurities people, it's ok to not understand some things and be open about it. Everyone doesn't understand more things than they do.
- Maxatar 2y ago
  Apart from one flagged comment I find the comments to be quite profound and interesting, we even have a cool visualization demo of the Riemann zeta function: https://news.ycombinator.com/item?id=40571995#40576767 https://news.ycombinator.com/item?id=40571995#40576767 Your comment is rather condescending and kind of feels like a form of projection rather than a meaningful contribution.
  - greenthrow 2y ago
    The comment you link is two hours newer than mine. I also didn't say it's only that type of comment. But when I commented, there were several comments that were just completely off base or bordering on nonsensical due to not understanding the subject. This is a toxic phenomenon that we don't really address, and I thought was interesting to highlight and also share a positive message to those afflicted by the malady of insecurity. You can choose to read in whatever you want, but that was (pretty obviously) my intent.
- CyberDildonics 2y ago
  This comments section is very oddly filled with people who don't actually understand the subject matter but want to sound smart, and then accomplish the opposite. Just this one?
- tucnak 2y ago
  Oh, I knew your username sounded familiar. Mister Smarty-Pansy-Cynical-McLoser back at it again!
breck 2y ago
MathWorld on Riemann: https://mathworld.wolfram.com/RiemannZetaFunction.html https://mathworld.wolfram.com/RiemannZetaFunction.html
blackle 2y ago
Relevant xkcd: https://xkcd.com/2595/ https://xkcd.com/2595/
- GalaxyNova 2y ago
  There's always a relevant xkcd isn't there
nickcw 2y ago
James Maynard appears regularly on Numberphile so if you'd like to hear some accessible mathematics from one of the authors of this paper I suggest you check it out: https://www.youtube.com/playlist?list=PLt5AfwLFPxWJdwkdjaK1ogByEGiVHdEnM https://www.youtube.com/playlist?list=PLt5AfwLFPxWJdwkdjaK1o...
- nomilk 2y ago
  TIL the Fields medal is only awarded to mathematicians under the age of 40. Source: https://www.youtube.com/watch?v=eupAXdWPvX8&list=PLt5AfwLFPxWJdwkdjaK1ogByEGiVHdEnM&index=3&t=10m https://www.youtube.com/watch?v=eupAXdWPvX8&list=PLt5AfwLFPx...
  - adtac 2y ago
    And only once every four years. Unfortunately, they award it on the 4n+2 years. As someone born on a 4n+0 year, I’ll have just 38 years, which is too severe a disadvantage for me to stomach, so I didn’t bother with it. 4n+2 people, you have no excuses.
    - gosub100 2y ago
      The margin is too thin, in other words?
      - manuelmoreale 2y ago
        What a great reference! Well done.
        arrowsmith 2y ago
        I don't get it.
        seanhunter 2y ago
        Pierre de Fermat wrote his "last theorem" (that no three positive integers a, b, c satisfy a^n + b^n = c^n for integer values of n>2) in the margin of a copy of Diophantus' arithmetica with the comment that "I have a truly marvelous proof which this margin is too thin to contain". The theorem went unproved for 385 years until Andrew Wiles eventually proved it in 1995, winning the Abel prize (because he was too old to win a Fields medal). Most people think Fermat didn't have a proof - just an intuition - as Wiles' proof uses a bunch of extremely sophisticated and powerful results that came well after Fermat. The GP's comment is particularly clever because until Wiles, the Riemann hypothesis and Fermat's last theorem were the two most famous unsolved problems in maths, so were often talked about together as being these almost unassailable challenges. https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem
samscully 2y ago
For anyone looking for an introduction to the Riemann Hypothesis that goes deeper than most videos but is still accessible to someone with a STEM degree I really enjoyed this video series [1] by zetamath. I understood everything in Profesor Tao's OP up to the part about "controlling a key matrix of phases" so the videos must have taught me something! [1] https://www.youtube.com/watch?v=oVaSA_b938U&list=PLbaA3qJlbE93DiTYMzl0XKnLn5df_QWqY https://www.youtube.com/watch?v=oVaSA_b938U&list=PLbaA3qJlbE...
amirhirsch 2y ago
I made this visualization of the zeta function in javascript, it is infinitely zoomable, and you can play around with parameters: https://amirhirsch.com/zeta/index.html https://amirhirsch.com/zeta/index.html It might help you understand why the hypothesis is probably true. It renders the partial sums and traces the path of zeta. In my rendering, I include all partial sums up to an automatically computed "N-critical" which is when the phase difference between two summands is less than pi (nyquist limit!), after which the behavior of the partial sums is monotonic. The clusters are like alias modes that go back and forth when the instantaneous frequency of the summands is between kpi and (k+1)pi, and the random walk section is where you only have one point per alias-mode. The green lines highlight a symmetry of the partial sums, where the clusters maintain symmetry with the random walk section, this symmetry is summarized pretty well in this paper: https://arxiv.org/pdf/1507.07631 https://arxiv.org/pdf/1507.07631
- atonalfreerider 2y ago
  Me too :) Mine is in Unity and shows the spiral in 3D, up the Y axis. I think it's helpful to see in three dimensions: https://github.com/atonalfreerider/riemann-zeta-visualization https://github.com/atonalfreerider/riemann-zeta-visualizatio...
- amirhirsch 2y ago
  I formed an intuitive signal processing interpretation of the Riemann Hypothesis many years ago, which I'll try to summarize briefly here. You can think of the Zeta function as a log-time sampler -- zeta(s) is the Laplace transform of sum(delta(t-ln n)) which samples at time t=(ln n) for integers n>0, a rapidly increasing sample rate. You can imagine this as an impulse response coming from a black box, and the impulse response can either be finite in energy or a power signal depending on the real parameter. If you suppose that the energy sum(|1/s|^2) is finite (ie real(s) > 1/2), then the Riemann Hypothesis implies that the sum is non-zero. It is akin to saying that the logarithmic sampler cannot destroy information without being plugged-in.
- 3abiton 2y ago
  Any good easily digestible sources on this topic?
brcmthrowaway 2y ago
Why does the universe require so much grinding for this problem It makes no sense if the rest of the universe relies upon simple rules
- gus_massa 2y ago
  There are some math problems with that are simple to ask but very difficult to solve. For example https://en.wikipedia.org/wiki/Four_color_theorem https://en.wikipedia.org/wiki/Four_color_theorem You need a long explanation to reduce the problem to only a thousand of cases, and then you can test the thousand of cases with a computer or write the solution of each case if you are brave enough. https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem You replace the original integer numbers with integer complex numbers and get a solution for small n. But for big n you realize some properties of factorization don't work anymore, and then have to write a few books of algebra and then understand the conection with eliptic curves (whatever they are [1]) and modular forms (whatever they are [2]) and then write another book just to prove the tricky part of the theorem. [1] I studies eliptic surves as a small part of a course, only for two or three weeks. I'm not sure how they are conected with this. [2] I have no idea what they are.
  - oulu2006 2y ago
    [1] neither [2] neither :) but I'm glad there are people out there that do!
billforsternz 2y ago
I think I read somewhere ("The Music of the Primes" maybe?) that the Riemann Hypothesis says that all zeroes of the Zeta function are on a line in the complex plane, and that although it is unproved, it has been proved all the zeroes are within a narrow strip centred on the line and you can make the strip as arbitrarily narrow as you like. I often wonder if I misunderstood somehow because it seems to me if it is really as I just restated it, well the Riemann Hypothesis is obviously true, that proof is "good enough" for engineering purposes anyway.
- gavagai691 2y ago
  "It has been proved all the zeroes are within a narrow strip centred on the line and you can make the strip as arbitrarily narrow as you like." Nothing close to this is known. The nontrivial zeros of zeta lie within the critical strip, i.e., 0 <= Re(s) <= 1 (in analytic number theory, the convention, going back to Riemann's paper is to write a complex variable as s = sigma + it)*. The Riemann Hypothesis states that all zeros of zeta are on the line Re(s) = 1/2. The functional equation implies that the zeros of zeta are symmetric about the line Re(s) = 1/2. Consequently, RH is equivalent to the assertion that zeta has no zeros for Re(s) > 1/2. A "zero-free region" is a region in the critical strip that is known to have no zeros of the Riemann zeta function. RH is equivalent to the assertion that Re(s) > 1/2 is a "zero-free region." The main reason that we care about RH is that RH would give essentially the best possible error term in the prime number theorem (PNT) https://en.wikipedia.org/wiki/Prime_number_theorem https://en.wikipedia.org/wiki/Prime_number_theorem. A weaker zero-free region gives a weaker error term in the PNT. The PNT in its weakest, ineffective form is equivalent to the assertion that Re(s) >= 1 is a zero free region (i.e., that there are no zeros on the line Re(s) = 1). The best-known zero-free region for zeta is the Vinogradov--Korobov zero-free region. This is the best explicit form of Vinogradov--Korobov known today https://arxiv.org/abs/2212.06867 https://arxiv.org/abs/2212.06867 (a slight improvement of https://arxiv.org/abs/1910.08205 https://arxiv.org/abs/1910.08205). I think your confusion stems from the fact that approximately the reverse of what you said above is true. That is, the best zero-free regions that we know get arbitrarily close to the Re(s) = 1 line (i.e., get increasingly "useless") as the imaginary part tends to infinity. Your statement seems to suggest that the the area we know contains the zeros gets arbitrarily close to the 1/2 line (which would be amazing). In other words, rather than our knowledge being about as close to RH as possible (as you suggested), our knowledge is about as weak as it could be. (See this image: https://commons.wikimedia.org/wiki/File:Zero-free_region_for_the_Riemann_zeta-function.svg https://commons.wikimedia.org/wiki/File:Zero-free_region_for.... The blue area is the zero-free region.) * I don't like this convention; why is it s = sigma + it instead of sigma = s + it? Blame Riemann.
idkdotcom 2y ago
[dead]
Havoc 2y ago
No idea what any of that means but they’re clearly stoked about it so happy for them
trojanhunt5 2y ago
[dead]
nsoonhui 2y ago
Fun fact: one of the author, Larry Guth, is the son of Alan Guth, theoretical physicist of inflation Cosmology fame (https://en.wikipedia.org/wiki/Larry_Guth https://en.wikipedia.org/wiki/Larry_Guth)
gtsnexp 2y ago
Is there any chance they made this talk available online?

540 points by pera 2y ago

ganzuul 2y ago

This improves modeling of long-tail distributions? How far off am I here?

gus_massa 2y ago
If the complex number is x+iy: They had a good bound of the long-tail distribution when x>=3/5=0.6. Now someone extended that result to x>=13/25=0.52. (The long term objective is to prove a stronger version for x>1/2=.5.)
- deleted 2y ago
  [deleted]
- konstantinua00 2y ago
  it's different affected distribution is always x>3/4, both before and after what's measured is upper bound on number of zeroes <y, relative to y it was <y^(3/5), now it's <y^(13/25) it says nothing about absence of zeroes, but the density result already affects prime distributions
  - gus_massa 2y ago
    My bad. Thanks for the correction.
lupire 2y ago
Can you be more specific? This is a result about a specific approximately known distribution.
- ganzuul 2y ago
  Excuse me I meant the related heavy-tailed distributions, which show up in dragon king theory. I'm wondering if this result indicates we have a new method to eke out important signals from noise that otherwise get smoothed out.

jl6 2y ago

Without intending to take anything away at all from the work of the researchers, I think “breakthrough” might just be overselling it. They have improved a bound but I can’t tell that their method will open up a plausible path to a full proof. It feels like we’ve managed to increase the top speed of a car from 200mph to 227mph, but are no closer to understanding how the engine works.

magnio 2y ago
Change that to CPU performance in 2023 and you got a breakthrough as well.
math_dandy 2y ago
It may take many breakthroughs to yield a proof of the Riemann Hypothesis. Progress on a notoriously hard problem through introduction of innovative new ideas can certainly qualify as a breakthrough, imo. Naturally, the importance of the breakthrough will be judged by what can be built upon it.
- fishtockos 2y ago
  This is clever application of existing ideas, not the sort of new idea which is needed. No one can really imagine what sort of machinery will need to be built to prove the RH. A proof is completely outside the realm of speculation at this point.
btilly 2y ago
When you've been stuck for 80 years, any improvement at all seems like a big deal. But more importantly, a new idea like this often points the way to variations that may also create improvement. Therefore as other researchers pile on, we may see this bound come down further. So yes, it is like a wall in a maze was breached, and there is the opportunity to go further. And it is fair to call that a breakthrough. Of course you're still charging into a maze, and further progress is far from guaranteed.
gus_massa 2y ago
The author of the post is Terence Tao that is the best live mathematician. If he says it's a "breakthrough", it's a breakthrough. (Anyway, I agree that if you are not in this research area, it's still an internal result that you can safetly ignore.) Also, in the last years he was leading a few project to crowsource improvements after breakthroughs in other conjetures. I'm too far from this area to be sure, but if he sees there is a chance to improve this, I expect him to launch a new one. From the article: > but they do a number of clever and unexpected maneuvers If he post something like that about me, I'd frame it an hang it in the wall.

pkilgore 2y ago

Found this[1] a useful primer on potential significance from a 2018 proposed proof. [1] https://www.sciencenews.org/article/why-we-care-riemann-hypothesis-math-prime-numbers https://www.sciencenews.org/article/why-we-care-riemann-hypo...

nomilk 2y ago
Also curious about the potential significance of a proof. The article is vague: > (primes) are important for securing encrypted transmissions sent over the internet. And importantly, a multitude of mathematical papers take the Riemann hypothesis as a given. If this foundational assumption were proved correct, “many results that are believed to be true will be known to be true,” says mathematician Ken Ono of Emory University in Atlanta. “It’s a kind of mathematical oracle.” Are there some obvious, known applications where a RH proof would have immediate practical effects? (beyond satisfaction and 'slightly better encryption').
- dgacmu 2y ago
  If we knew that the extended Riemann hypothesis was true, we could use the Miller test for deterministic primality testing in log(n)^4; the AKS test, which doesn't depend on RH, is lg(n)^6. Do we care? Not for most applications -- doing a bunch of randomized Miller-Rabin tests is fine for most practical purposes. But it would be really nice to have a faster deterministic algorithm around. AKS isn't practical for anything; miller... Miiiiiggghtt be.
  - olddustytrail 2y ago
    Then why not just assume it's true and if something breaks, hey, you might just have disproved it?
    - tekchip 2y ago
      [flagged]
    - aj7 2y ago

greekanalyst 2y ago

Always amazed by Terence Tao's clarity of thought. Not only is he one of the greatest mathematical minds alive (if not the greatest), he is also one of the most eloquent writers on mathematics. Nothing more beautiful than seeing great science being married with great writing. Even if you don't understand the specifics, you can always get the big picture. Thank you, Terence!

alexander2002 2y ago
Does he have any beginner friendly books on math I saw some clips of his masterclass but I am skeptical of masterclass in general so I am not sure about that
- dan-robertson 2y ago
  He has a blog which has some more elementary posts and some less. But I’m not sure if that would be sufficiently ‘beginner friendly’ for your needs. https://terrytao.wordpress.com/ https://terrytao.wordpress.com/
- ted_dunning 2y ago
  His masterclass is substantially better than the average masterclass, particularly for somebody who isn't already an amateur or better mathematician.
- empeyot 2y ago
  His book Analysis I, and potentially also his linear algebra lecture come to mind.

idlewords 2y ago

Trying to imagine what it must feel like to have Terence Tao summarize your argument while mentioning that he'd tried something similar but failed. "The arguments are largely Fourier analytic in nature. The first few steps are standard, and many analytic number theorists, including myself, who have attempted to break the Ingham bound, will recognize them; but they do a number of clever and unexpected maneuvers."

mi_lk 2y ago
I mean, two authors of the paper are pretty well established in the field already [0]: https://en.wikipedia.org/wiki/Larry_Guth https://en.wikipedia.org/wiki/Larry_Guth [1]: https://en.wikipedia.org/wiki/James_Maynard_(mathematician) https://en.wikipedia.org/wiki/James_Maynard_(mathematician)
- chx 2y ago
  Yes but it's Terence Tao. I mean, the set of living mathematicians is not well ordered on greatness but if it were, Terence Tao would be fairly close to the upper limit.
  - Ar-Curunir 2y ago
    Maynard is a Fields Medallist, so is also one of the strongest mathematicians around.
  - alkyon 2y ago
    Maynard won a Fields medal, Tao is of course in the elite, but so is JM.
deleted 2y ago
[deleted]
nybsjytm 2y ago
It's perfectly common for a mathematician to successfully use a technique where another top mathematician tried and failed.

mr-roboto 2y ago

Good timing for me. I'm in the middle of listening to "The Humans" by Matt Haig. The story opens after someone proves the Riemann hypothesis.

vsnf 2y ago

What are the practical implications of a conclusion of the hypothesis, in either direction?

throw_pm23 2y ago
This is pure math, so not much, at least directly or immediately. But I find it amusing how this argument always comes up and how it goes back millenia. A student of Plato (428 B.C. -- 348 B.C.) once asked the great master, "What practical uses do these theorems serve? What is to be gained from them?" Plato's answer was immediate and peremptory. He turned to one of his slaves and said, "Give this young man an obol [a small Greek coin] so that he may feel that he has gained something from my teachings. Then expel him."
- vsnf 2y ago
  I didn’t mean to imply that it needed to be useful. I just hear about the hypothesis a lot and I wonder what the immediate knock-on effects would be. Does it unlock other theoretical work, are there other proofs that work or don’t work depending on it, etc. And if it has an effect on something real or tangible that’s even better.
  - moi2388 2y ago
    In principle a proof of the Riemann Hypothesis could give you information about the distribution of primes and could possible make it easier to test for primality, in the worst case breaking modern encryption. But that’s a lot of what ifs away.
    - Ar-Curunir 2y ago
      There is no known result that says RH being true breaks modern encryption; if there was, the cryptanalysts would assume RH and try to break it anyway.
      - IsTom 2y ago

wing-_-nuts 2y ago

Can I get a ELI!M[Not a mathematician]?

nhatcher 2y ago
One of the most important open problems in mathematics is called the Riemann hypothesis. It states that the solutions of a certain equation `zeta(z)=0` are all of a particular type. Almost every living mathematician has tried to solve it at some point in their lives. The implications of the hypothesis are deep for the theory of numbers, for instance for the distribution of prime numbers. In a recent paper some mathematicians claim they have put some stronger bounds on where those solutions can be. In this link Terrence Tao, one of the most acclaimed mathematicians alive speaks very highly about the paper. IMHO, this is probably not of huge interest to not mathematicians just yet. It is an extremely technical result. And pending further review it might very well be wrong or incomplete. There are lots of places you can read about the Riemann Hypothesis, its implications and its attempts to solve it.
- chii 2y ago
  As for why the Riemann Hypothesis itself is interesting, it is because this zeta function seems to have information about the primes inside it. The eye opener for me is when a video about it connected the idea of the zeta function to how you might create a square wave by adding successive sine waves (of higher and higher frequency). Imagine a function (call it `prime_count`), which gives you the number of prime numbers below it's argument. If you tried to plot this function on a graph, it will look like a series of jaggard steps - a jump when you reach a new prime number. It turns out, if you rewrite the zeta function and decompose it (into it's infinitely many zeros - akin to doing a "taylor series" expansion for other functions), you will find that the more zeros you use, the closer you will get to a jaggard graph. This video is really good at giving a visual explanation : https://youtu.be/e4kOh7qlsM4?t=594 https://youtu.be/e4kOh7qlsM4?t=594
  - seanhunter 2y ago
    > The eye opener for me is when a video about it connected the idea of the zeta function to how you might create a square wave by adding successive sine waves (of higher and higher frequency). The process of taking an arbitrary waveform and approximating it by adding together a series of sine and/or cosine waves with different amplitudes etc is called a Fourier transform. That is what Terrence Tao is talking about when he mentions trying this approach using Fourier analysis himself. [1] Fourier transforms are used all over the place, eg the discrete cosine transform (DCT) is part of JPEG, MP3 etc. [1] https://en.wikipedia.org/wiki/Fourier_analysis https://en.wikipedia.org/wiki/Fourier_analysis

pheatherlite 2y ago

[flagged]

ipnon 2y ago
I like how Tao can just write a few quick candid thoughts and post them online for everyone.
jcla1 2y ago
I think giving any more details than his three short posts would delve too far into the specific arguments of the paper to be valuable for a wide enough audience, subtracting those that are going to be looking at the pre-print anyhow.
rrauenza 2y ago
I have discovered a truly marvelous demonstration of this proposition that .... will not fit in 280 characters. -- Fermat
deleted 2y ago
[deleted]

cvoss 2y ago

What are your opinions of all the theorems that rely on RH as an excluded middle? Constructivists reject exmid, saying instead that a proof of "A or B" requires you to have in hand a proof of A or a proof of B. And nobody yet has a proof of RH nor a proof of ~RH. This is important in so-called incomplete logical systems, where some theorems are neither provable nor disprovable, and, therefore, exmid is an inadmissible axiom.

nerdponx 2y ago
Aren't those sort of different things? I thought the whole point of provability was that it was distinct from truth.
- floxy 2y ago
  How do you know something is true if you don't have a proof? It all depends on you views on the philosophy of mathematics. Are there "true" statements that don't have proofs? Some say yes, there are platonic ideas that are true, even if they aren't provable. Others say, "what does it even mean to say something is true, if there is no proof. What you really have is a conjecture."
  - lisper 2y ago
    There is another possibility: it could be an arbitrary choice, as in the case of the parallel postulate, the axiom of choice, and non-standard models of the Peano axioms.
    - btilly 2y ago
      According to logicians, it always is an arbitrary choice. But we have a second-order notion of what "finite integer" should mean. And within that notion the idea might be false. Here's how that plays out. Suppose that the RH cannot be proved or disproved from ZF. (It turns out that choice cannot matter for all theorems in number theory, so ZF is enough.) That means that there is a model of ZF in which RH is true. Every model of ZF contains a finite calculation for any non-trivial zero of the Riemann function. (The word "finite" is doing some heavy lifting here - it is a second order concept.) That calculation must work out the same in all models. Therefore every finite nontrivial zero has complex part 0.5. Therefore RH is actually true of the standard model of the integers. Therefore every model of ZF where RH is false, is non-standard. The truth of RH is therefore independent of ZF. But it's true in our second order notion of what the integers are.

greenthrow 2y ago

This comments section is very oddly filled with people who don't actually understand the subject matter but want to sound smart, and then accomplish the opposite. Let go of your insecurities people, it's ok to not understand some things and be open about it. Everyone doesn't understand more things than they do.

Maxatar 2y ago
Apart from one flagged comment I find the comments to be quite profound and interesting, we even have a cool visualization demo of the Riemann zeta function: https://news.ycombinator.com/item?id=40571995#40576767 https://news.ycombinator.com/item?id=40571995#40576767 Your comment is rather condescending and kind of feels like a form of projection rather than a meaningful contribution.
- greenthrow 2y ago
  The comment you link is two hours newer than mine. I also didn't say it's only that type of comment. But when I commented, there were several comments that were just completely off base or bordering on nonsensical due to not understanding the subject. This is a toxic phenomenon that we don't really address, and I thought was interesting to highlight and also share a positive message to those afflicted by the malady of insecurity. You can choose to read in whatever you want, but that was (pretty obviously) my intent.
CyberDildonics 2y ago
This comments section is very oddly filled with people who don't actually understand the subject matter but want to sound smart, and then accomplish the opposite. Just this one?
tucnak 2y ago
Oh, I knew your username sounded familiar. Mister Smarty-Pansy-Cynical-McLoser back at it again!

breck 2y ago

MathWorld on Riemann: https://mathworld.wolfram.com/RiemannZetaFunction.html https://mathworld.wolfram.com/RiemannZetaFunction.html

blackle 2y ago

Relevant xkcd: https://xkcd.com/2595/ https://xkcd.com/2595/

GalaxyNova 2y ago
There's always a relevant xkcd isn't there

nickcw 2y ago

James Maynard appears regularly on Numberphile so if you'd like to hear some accessible mathematics from one of the authors of this paper I suggest you check it out: https://www.youtube.com/playlist?list=PLt5AfwLFPxWJdwkdjaK1ogByEGiVHdEnM https://www.youtube.com/playlist?list=PLt5AfwLFPxWJdwkdjaK1o...

nomilk 2y ago
TIL the Fields medal is only awarded to mathematicians under the age of 40. Source: https://www.youtube.com/watch?v=eupAXdWPvX8&list=PLt5AfwLFPxWJdwkdjaK1ogByEGiVHdEnM&index=3&t=10m https://www.youtube.com/watch?v=eupAXdWPvX8&list=PLt5AfwLFPx...
- adtac 2y ago
  And only once every four years. Unfortunately, they award it on the 4n+2 years. As someone born on a 4n+0 year, I’ll have just 38 years, which is too severe a disadvantage for me to stomach, so I didn’t bother with it. 4n+2 people, you have no excuses.
  - gosub100 2y ago
    The margin is too thin, in other words?
    - manuelmoreale 2y ago
      What a great reference! Well done.
      - arrowsmith 2y ago
        I don't get it.
        seanhunter 2y ago
        Pierre de Fermat wrote his "last theorem" (that no three positive integers a, b, c satisfy a^n + b^n = c^n for integer values of n>2) in the margin of a copy of Diophantus' arithmetica with the comment that "I have a truly marvelous proof which this margin is too thin to contain". The theorem went unproved for 385 years until Andrew Wiles eventually proved it in 1995, winning the Abel prize (because he was too old to win a Fields medal). Most people think Fermat didn't have a proof - just an intuition - as Wiles' proof uses a bunch of extremely sophisticated and powerful results that came well after Fermat. The GP's comment is particularly clever because until Wiles, the Riemann hypothesis and Fermat's last theorem were the two most famous unsolved problems in maths, so were often talked about together as being these almost unassailable challenges. https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem

samscully 2y ago

For anyone looking for an introduction to the Riemann Hypothesis that goes deeper than most videos but is still accessible to someone with a STEM degree I really enjoyed this video series [1] by zetamath. I understood everything in Profesor Tao's OP up to the part about "controlling a key matrix of phases" so the videos must have taught me something! [1] https://www.youtube.com/watch?v=oVaSA_b938U&list=PLbaA3qJlbE93DiTYMzl0XKnLn5df_QWqY https://www.youtube.com/watch?v=oVaSA_b938U&list=PLbaA3qJlbE...

amirhirsch 2y ago

I made this visualization of the zeta function in javascript, it is infinitely zoomable, and you can play around with parameters: https://amirhirsch.com/zeta/index.html https://amirhirsch.com/zeta/index.html It might help you understand why the hypothesis is probably true. It renders the partial sums and traces the path of zeta. In my rendering, I include all partial sums up to an automatically computed "N-critical" which is when the phase difference between two summands is less than pi (nyquist limit!), after which the behavior of the partial sums is monotonic. The clusters are like alias modes that go back and forth when the instantaneous frequency of the summands is between kpi and (k+1)pi, and the random walk section is where you only have one point per alias-mode. The green lines highlight a symmetry of the partial sums, where the clusters maintain symmetry with the random walk section, this symmetry is summarized pretty well in this paper: https://arxiv.org/pdf/1507.07631 https://arxiv.org/pdf/1507.07631

atonalfreerider 2y ago
Me too :) Mine is in Unity and shows the spiral in 3D, up the Y axis. I think it's helpful to see in three dimensions: https://github.com/atonalfreerider/riemann-zeta-visualization https://github.com/atonalfreerider/riemann-zeta-visualizatio...
amirhirsch 2y ago
I formed an intuitive signal processing interpretation of the Riemann Hypothesis many years ago, which I'll try to summarize briefly here. You can think of the Zeta function as a log-time sampler -- zeta(s) is the Laplace transform of sum(delta(t-ln n)) which samples at time t=(ln n) for integers n>0, a rapidly increasing sample rate. You can imagine this as an impulse response coming from a black box, and the impulse response can either be finite in energy or a power signal depending on the real parameter. If you suppose that the energy sum(|1/s|^2) is finite (ie real(s) > 1/2), then the Riemann Hypothesis implies that the sum is non-zero. It is akin to saying that the logarithmic sampler cannot destroy information without being plugged-in.
3abiton 2y ago
Any good easily digestible sources on this topic?

brcmthrowaway 2y ago

Why does the universe require so much grinding for this problem It makes no sense if the rest of the universe relies upon simple rules

gus_massa 2y ago
There are some math problems with that are simple to ask but very difficult to solve. For example https://en.wikipedia.org/wiki/Four_color_theorem https://en.wikipedia.org/wiki/Four_color_theorem You need a long explanation to reduce the problem to only a thousand of cases, and then you can test the thousand of cases with a computer or write the solution of each case if you are brave enough. https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem https://en.wikipedia.org/wiki/Fermat%27s_Last_Theorem You replace the original integer numbers with integer complex numbers and get a solution for small n. But for big n you realize some properties of factorization don't work anymore, and then have to write a few books of algebra and then understand the conection with eliptic curves (whatever they are [1]) and modular forms (whatever they are [2]) and then write another book just to prove the tricky part of the theorem. [1] I studies eliptic surves as a small part of a course, only for two or three weeks. I'm not sure how they are conected with this. [2] I have no idea what they are.
- oulu2006 2y ago
  [1] neither [2] neither :) but I'm glad there are people out there that do!

billforsternz 2y ago

I think I read somewhere ("The Music of the Primes" maybe?) that the Riemann Hypothesis says that all zeroes of the Zeta function are on a line in the complex plane, and that although it is unproved, it has been proved all the zeroes are within a narrow strip centred on the line and you can make the strip as arbitrarily narrow as you like. I often wonder if I misunderstood somehow because it seems to me if it is really as I just restated it, well the Riemann Hypothesis is obviously true, that proof is "good enough" for engineering purposes anyway.

gavagai691 2y ago
"It has been proved all the zeroes are within a narrow strip centred on the line and you can make the strip as arbitrarily narrow as you like." Nothing close to this is known. The nontrivial zeros of zeta lie within the critical strip, i.e., 0 <= Re(s) <= 1 (in analytic number theory, the convention, going back to Riemann's paper is to write a complex variable as s = sigma + it)*. The Riemann Hypothesis states that all zeros of zeta are on the line Re(s) = 1/2. The functional equation implies that the zeros of zeta are symmetric about the line Re(s) = 1/2. Consequently, RH is equivalent to the assertion that zeta has no zeros for Re(s) > 1/2. A "zero-free region" is a region in the critical strip that is known to have no zeros of the Riemann zeta function. RH is equivalent to the assertion that Re(s) > 1/2 is a "zero-free region." The main reason that we care about RH is that RH would give essentially the best possible error term in the prime number theorem (PNT) https://en.wikipedia.org/wiki/Prime_number_theorem https://en.wikipedia.org/wiki/Prime_number_theorem. A weaker zero-free region gives a weaker error term in the PNT. The PNT in its weakest, ineffective form is equivalent to the assertion that Re(s) >= 1 is a zero free region (i.e., that there are no zeros on the line Re(s) = 1). The best-known zero-free region for zeta is the Vinogradov--Korobov zero-free region. This is the best explicit form of Vinogradov--Korobov known today https://arxiv.org/abs/2212.06867 https://arxiv.org/abs/2212.06867 (a slight improvement of https://arxiv.org/abs/1910.08205 https://arxiv.org/abs/1910.08205). I think your confusion stems from the fact that approximately the reverse of what you said above is true. That is, the best zero-free regions that we know get arbitrarily close to the Re(s) = 1 line (i.e., get increasingly "useless") as the imaginary part tends to infinity. Your statement seems to suggest that the the area we know contains the zeros gets arbitrarily close to the 1/2 line (which would be amazing). In other words, rather than our knowledge being about as close to RH as possible (as you suggested), our knowledge is about as weak as it could be. (See this image: https://commons.wikimedia.org/wiki/File:Zero-free_region_for_the_Riemann_zeta-function.svg https://commons.wikimedia.org/wiki/File:Zero-free_region_for.... The blue area is the zero-free region.) * I don't like this convention; why is it s = sigma + it instead of sigma = s + it? Blame Riemann.

idkdotcom 2y ago

[dead]

Havoc 2y ago

No idea what any of that means but they’re clearly stoked about it so happy for them

trojanhunt5 2y ago

[dead]

nsoonhui 2y ago

Fun fact: one of the author, Larry Guth, is the son of Alan Guth, theoretical physicist of inflation Cosmology fame (https://en.wikipedia.org/wiki/Larry_Guth https://en.wikipedia.org/wiki/Larry_Guth)

gtsnexp 2y ago

Is there any chance they made this talk available online?