[HN Gopher] A breakthrough towards the Riemann hypothesis
___________________________________________________________________
A breakthrough towards the Riemann hypothesis
Author : pera
Score : 526 points
Date : 2024-06-04 08:25 UTC (1 days ago)
(HTM) web link (mathstodon.xyz)
(TXT) w3m dump (mathstodon.xyz)
| ganzuul wrote:
| This improves modeling of long-tail distributions? How far off am
| I here?
| gus_massa wrote:
| 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.)
| konstantinua00 wrote:
| 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 wrote:
| My bad. Thanks for the correction.
| lupire wrote:
| Can you be more specific? This is a result about a specific
| approximately known distribution.
| ganzuul wrote:
| 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 wrote:
| 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 wrote:
| Change that to CPU performance in 2023 and you got a
| breakthrough as well.
| math_dandy wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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.
| jlarcombe wrote:
| Well, the two authors of the paper he's talking about are
| also amongst the foremost mathematicians in the world today.
| James Maynard won the Fields medal a couple of years ago. So
| they've probably got used to approbation from the likes of
| Terence Tao. Nonetheless, exciting!
| nybsjytm wrote:
| > 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.
|
| I think it's pretty silly to say that Tao is the best living
| mathematician, but even if he were I don't think that this
| would be a useful way to think about things.
| kadoban wrote:
| Yeah, random people on HN are much better judges of
| importance of mathematical results.
| gus_massa wrote:
| > _I think it 's pretty silly to say that Tao is the best
| living mathematician_
|
| I agree. I was going to write "one of the best" or
| "probably the best" or something like that. It's like
| discussing if Messi or Maradona were the best football
| players. [1]. Anyway, all three of them are quite good.
|
| > _but even if he were I don 't think that this would be a
| useful way to think about things._
|
| I also agree. It's just and argument by authority. For a
| decition that would change the lives of millons of persons
| and has a lot of subtle tradeoff and unknown unknowns, I'd
| ask for a better justification and evidence. But deciding
| if this is a breakthrough or not, may only change the lives
| of a few graduate students and a few grown up
| mathematicians, so I'm happy to take the word of Tao.
|
| [1] Hi from Argentina!
| btilly wrote:
| No matter what you think, best living mathematician is what
| other mathematicians say about him.
|
| But I'll humor you. How would you prefer we say things?
|
| - Highest IQ on record.
|
| - One of only 3 people to score over 700 on the Math SAT
| before he was 8. He had the highest score of the three with
| 760.
|
| - At ages 10, 11, and 12 he set the record for winning
| Bronze, Silver, and Gold respectively in the International
| Math Olympiad. After that he lost interest.
|
| - PHD from Princeton at 21.
|
| - Full professor at UCLA at 24. This is a record.
|
| - He is a respected leader in at least a half-dozen areas
| of mathematics. He regularly publishes in many more. It is
| unusual for a mathematician to have significant
| publications in 2 areas.
|
| - Wikipedia lists 28 major prizes for him. Excepting
| Australian of the Year in 2022, all are major mathematics
| prizes. No other mathematician comes close.
|
| - Once you exclude junk journals, Tao publishes papers
| faster than any other mathematician. And his are all good.
|
| - Tao's papers show up in the popular press more often than
| the next 3 mathematicians combined.
|
| And so on.
|
| At what point is there a better way to think about this
| than, "best live mathematician"?
|
| (And yeah, until I began noticing Tao, I would have also
| thought that a silly way to think...)
| fishtockos wrote:
| The idea that Tao has accomplished more than, say, Serre
| because the latter, who won the Fields medal at 27, only
| received his PhD at 25 and his bachelor's at 22 while the
| former received his PhD at 21 and his bachelor's at 16 is
| so absurd that it can be refuted merely by alluding to
| it.
|
| Your other points are similar.
| btilly wrote:
| Serre is indeed a top mathematician. (I'm actually
| surprised to find out that he's still alive!)
|
| At this point Tao only has 3/4 his number of
| publications, similar numbers of textbooks, a similar
| number of awards (using https://mathshistory.st-
| andrews.ac.uk/Biographies/Serre/ to count awards), and so
| on. I'd count Tao as having more of what I see as major
| breakthroughs, but that is subjective. But then again,
| Tao is half of Serre's age.
|
| Yeah. I still think it is fair to put Tao in the same
| tier as Euler, Gauss and Hilbert.
| auntienomen wrote:
| Time will tell.
| nybsjytm wrote:
| Sorry, I think this style of hagiography is completely
| goofy, there's nothing else to say about it. And I'm sure
| it's not even true that he has the "highest IQ on
| record."
|
| To make a claim like "greatest living mathematician" it
| would be more appropriate to talk about his actual
| research accomplishments and how they compare to
| canonical figures like Gromov, Milnor, Serre, Smale, Yau,
| among many younger counterparts.
|
| But, personally, I think defending any such claim for any
| particular person is kind of a juvenile exercise. I am a
| mathematician and among the slight minority of
| mathematicians I know who would take the question
| seriously, a minority of them would select Tao. Which of
| course isn't to say that he isn't universally considered
| a top mathematician.
| abetusk wrote:
| I'm no expert and I don't like engaging in arguments to
| authority but Terrence Tao is one of the leading mathematicians
| of our age, akin to Erdos. When he's excited about a result,
| that should be a high signal indicator that it's worth paying
| attention.
|
| In terms of the result itself, as Tao explains:
|
| > ... making the first substantial improvement to a classical
| 1940 ...
|
| Meaning, it's been 80 years since any progress has been made on
| this front (bounds of zeros of the Riemann zeta function).
|
| I would make the argument that it's not so much making a top
| speed car from 200mph to 227mph but discovering an engine that
| uses gasoline instead of steam.
|
| Presumably the methods used in the paper might be able to be
| extended or refined to get better results.
| QuesnayJr wrote:
| It probably doesn't open up a path to a full proof, simply
| because Guth and Maynard are excellent mathematicians, and they
| wouldn't have given away their approach yet if they thought
| they could prove the full thing.
|
| It's still important because there hasn't been much new
| evidence in years whether the conjecture is true or false. Now
| we have some new evidence that it's true.
| JadeNB wrote:
| I don't think that's really conclusive; surely they've taken
| it as far as they think they can in a reasonable amount of
| time, but (1) they've no real need to be secretive--why not
| share it as soon as there's a concrete result?, and (2)
| sometimes the inventor of a technique can be too used to the
| old way of thinking to use their new technique to its full
| potential, and it takes less experienced eyes on it.
| lupire wrote:
| That's not how math community generally works.
|
| Everyone knows what the important idea of a proof is.
| Publishing the work that grates a new field of mathematics
| that leads to a proof is not worse than toiling away
| privately (for longer!).
|
| Gregory Perelman, the only Millennium Prize winner, declined
| the prize because he refused to accept credit for Richard S
| Hamilton's foundational work.
| pkilgore wrote:
| Found this[1] a useful primer on potential significance from a
| 2018 proposed proof.
|
| [1] https://www.sciencenews.org/article/why-we-care-riemann-
| hypo...
| nomilk wrote:
| 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 wrote:
| 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 wrote:
| Then why not just assume it's true and if something breaks,
| hey, you might just have disproved it?
| aj7 wrote:
| What if an airliner or a spacecraft or a self- driving
| car or a surgical robot breaks?
| olddustytrail wrote:
| They already do. Was this supposed to be an enlightening
| question?
| pfdietz wrote:
| The randomized algorithm doesn't need to be perfect, it
| just needs to present a failure probability << the
| failure rate from other causes. And it can easily do
| that, since the failure probability goes down
| exponentially with the number of iterations of the test.
| dgacmu wrote:
| Because for practical things we have a faster solution
| that might also be wrong occasionally (randomized
| testing). There's no benefit to using any of the known
| deterministic tests if they're not truly deterministic.
| The speed gap vs randomized is huge.
| olddustytrail wrote:
| Then it seems it wouldn't be better to use the
| deterministic test ever. Just use the randomised test and
| accept the failure rate.
| tzs wrote:
| To put some numbers on it, assume a civilization of 1
| trillion people, with each person using 1 000 things that
| needs a large prime, and those 1 000 things need to
| generate a new prime 1 000 times a second.
|
| The most conservative bound on the chance that a random
| single Miller-Rabin test will say that a composite number
| is prime is 1/4.
|
| Using that bound, if that civilization used Miller-Rabin
| with 48 random Miller-Rabin tests to find primes they would
| get a composite falsely identified as a prime about once
| every 60 000 years.
|
| If they used 64 tests they would have one false positive
| about ever 258 trillion years. That's past the point when
| all stars in the universe have run out of fuel.
|
| Now assume that every star in the universe has a similar
| civilization. If everyone uses 96 Miller-Rabin tests there
| will be a false positive about once per 24 billion years.
|
| As I said that is using a false positive rate of 1/4, which
| is very conservative. There's a paper [1], "Average Case
| Error Estimates For the Strong Prime Test" by Damgard,
| Landrock, and Pomerance that gives several bounds.
|
| Their bounds are in terms of k, the number of bits of the
| odd number being tests, and t, the number of random Miller-
| Rabin tests. They give 4 bounds, for various ranges of k
| and t: (* Valid for k >= 2 *) k^2
| 4^(2-Sqrt[k]) (* Valid if t = 2, k >= 88 or if 3
| <= t <= k/9, k >= 21 *) k^(3/2) 2^t t^(-1/2)
| 4^(2-Sqrt[t k]) (* Valid for t >= k/9, k >= 21
| *) 7/20 k 2^(-5t) + 1/7 k^(15/4) 2^(-k/2-2t) + 12 k
| 2^(-k/4-3t) (* Valid for t >= k/4, k >= 21 *)
| 1/7 k^(15/4) 2^(-k/2-2t)
|
| It is the second one that is most relevant in most
| situations where you want a large prime and want to do as
| few tests as possible to get your false positive chances
| below your acceptable threshold.
|
| Using the hypothetical trillion being civilization with
| each being needing a million primes a second, here's the
| expected number of years between that civilization seeing a
| false positive using the second bound above, for k = 64 and
| t 2 through 5: 2 3.6 x 10^26 3 7.6 x
| 10^27 4 8.5 x 10^28 5 6.5 x 10^29
|
| The bound gets lower as k increases. If they needed 1024
| bit primes that table becomes: 2 1.5 x
| 10^45 3 1.3 x 10^51 4 1.1 x 10^56 5 2.1 x
| 10^60
|
| [1] https://math.dartmouth.edu/~carlp/PDF/paper88.pdf
| dgacmu wrote:
| Oh, no disagreement. "Practical" really means "practical
| for math things where you want to guarantee correctness,
| not just have extremely high confidence". I can't think
| of any normal use of primes that needs something past
| what you can do with a bunch of M-R tests.
|
| But we _do_ use a lot of computational power to verify
| finding particularly interesting primes, etc., so I
| maintain that it'd be a nice thing to have in our pocket.
| You never know when you'll find yourself on a deserted
| island with a powerful computer and need to
| deterministically prove the primality of a large number.
| ;)
| sdenton4 wrote:
| Of course, if you end up with an incorrect result due to
| assuming the Riemann hypothesis, you'll get a fields
| medal anyway...
| tzs wrote:
| Note: the first section, using the conservative estimate,
| is correct. The second using the tighter bounds is not. I
| multiplied where I should have divided, leading to an
| error of a factor or around 10^48 in both tables. E.g.,
| for generating 1024 bit primes with 5 tests the trillion
| people civilization generating a million primes a second
| per person would see one error around every 10^12 years,
| not one ever 10^60 years.
|
| For 64 bit primes they would need to do around 48 tests
| per prime to get down to a similar error rate.
|
| For
| zem wrote:
| as nomilk said above, for any practical real-world
| application you could likely just assume the reimann
| hypothesis was true and proceed accordingly
| spenczar5 wrote:
| In the realm of applications, most engineers would say that
| our confidence in the RH (or more realistically, downstream
| consequences thereof) is high enough to treat it as true. The
| proof is, for applications, a formality (albeit a wide-
| ranging, consequential one!).
|
| More likely, a proof of the Riemann Hypothesis would require
| new ideas, techniques, and math. It is probable that those
| devices would have broader reach.
|
| The applications of expansions in math often work that way:
| as we forge through the jungle, the tools we develop to make
| our way through are more useful than the destination.
| mtlmtlmtlmtl wrote:
| Mathematics is sort of strange in this regard in that there's
| lots of work already done that assumes RH, so many of the
| consequences of the theorem itself are already worked out.
| And RH seems to be true on extensive numerical searches(no
| counterexamples found). So the theorem being true wouldn't be
| earth-shattering in and of itself.
|
| It's more about the method used to prove the theorem, which
| might involve novel mathematics. And probably will in this
| case considering how long it's taking to prove RH. Since this
| method hasn't been found yet, it's hard to say what the
| consequences of it might be.
|
| At least that's my layman's understand of it.
| greekanalyst wrote:
| 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 wrote:
| 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 wrote:
| 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/
| ted_dunning wrote:
| His masterclass is substantially better than the average
| masterclass, particularly for somebody who isn't already an
| amateur or better mathematician.
| empeyot wrote:
| His book Analysis I, and potentially also his linear algebra
| lecture come to mind.
| idlewords wrote:
| 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 wrote:
| I mean, two authors of the paper are pretty well established in
| the field already
|
| [0]: https://en.wikipedia.org/wiki/Larry_Guth
|
| [1]:
| https://en.wikipedia.org/wiki/James_Maynard_(mathematician)
| chx wrote:
| 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 wrote:
| Maynard is a Fields Medallist, so is also one of the
| strongest mathematicians around.
| alkyon wrote:
| Maynard won a Fields medal, Tao is of course in the elite,
| but so is JM.
| nybsjytm wrote:
| It's perfectly common for a mathematician to successfully use a
| technique where another top mathematician tried and failed.
| random3 wrote:
| It must feel like meritocracy. Like when ranking, particularly
| in strict order, is not the norm - so Terrence Tao doesn't see
| himself "on top" of anything. Moreover it must imply some solid
| grounding and a good understanding of how someone's actions are
| not expected to be correlated with their reputation. This is
| especially the case where getting the results is a personal or
| strictly team effort, not a popularity contest.
|
| It can be unexpeted for anyone that's operating in the regular
| business, corporate, VC and general academic landscape where
| politics rule while meritocracy is a feel good motivator while
| popularity is the real coin.
| Ar-Curunir wrote:
| Tao and Maynard are both academics...
| ants_everywhere wrote:
| I haven't met him personally, but Tao's writing is very humble
| and very kind. He talks openly about trying things and not
| having them work out. And he writes in general a lot about
| tools and their limitations. I definitely recommend reading his
| blog.
| simple_quest_9 wrote:
| I find that mathematicians tend to have the smallest egos --
| eccentric as they may be. I think it's because the difficulty
| of mathematics reminds one of their fallibility.
|
| In school, I typically found the math and physics teachers to
| be humbler than the others. Not always, but, I couldn't help
| but notice that trend.
| mr-roboto wrote:
| 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 wrote:
| What are the practical implications of a conclusion of the
| hypothesis, in either direction?
| throw_pm23 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| 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 wrote:
| Not RH being true, but that the _proof_ itself would
| require discovering and proving something of high
| interest as an intermediate step.
| Ar-Curunir wrote:
| Perhaps, but it's not clear that would be
| cryptographically relevant still.
| Ar-Curunir wrote:
| Testing primality doesn't break encryption; we already
| test for primarily on a daily basis very efficiently
| moi2388 wrote:
| My apologies, that is not what I meant. I meant that the
| proof might break encryption. Again, _might_. As response
| to a list of _possible_ real-world implications as the
| reader asked for.
| kristopolous wrote:
| These things are kind of like landing on the moon. The act
| of actually walking on the moon is fairly symbolic and
| ultimately the more important things were the achievements
| required to get there.
|
| What the current state of the problem demonstrates is
| there's some unknown quantity of cutting edge work required
| that has yet to be done; maybe there's a cunning conceptual
| error in some field that has somehow missed everyone's eye
| or maybe there's an entirely new mathematical concept or
| tool which makes the problem straight-forward, but nobody
| has "invented it" yet.
|
| One day, the time we are in right now will be "150 years
| ago". Contemporaries of any time think all fundamental
| things have been exhausted and discovered for whatever
| reason. It was as true in 1874 as 2024.
|
| And they've always been wrong. Doesn't mean you or I can
| get there. It'll take some brilliant individual or team
| years of sweat equity, skill and profound luck but one day,
| right now too will be 150 years ago and people will look
| back on us as we look back on 1874 mathematics.
|
| Problems like Riemann is how we forge ahead.
| lupire wrote:
| There's another possibility: that everything
| _discoverable_ by humans as been discovered.
|
| Before long, AIs and organizations will be claiming to
| prove things that no human has the capacity to verify
| with reasonable confidence, or humans will run or of
| capacity or make progress. We just had the 1000 page
| Geometric Langlands proof.
| kristopolous wrote:
| "everything discoverable by humans as been discovered"
| again, this claim is made by every generation like
| clockwork, going back as long as humans have been making
| claims that can be recorded.
|
| I trust it's as wrong now as it always has been.
| bonzini wrote:
| https://en.wikipedia.org/wiki/Riemann_hypothesis#Consequence...
| lists several theorems that have been proved conditionally on
| the Riemann hypothesis being true. The most notable is probably
| the Goldbach conjecture, though that requires the generalized
| Riemann hypothesis.
| Maxatar wrote:
| It's a weak form of the Goldbach conjecture not the actual
| Goldbach conjecture itself. The weak form also has a proof
| that does not depend on the Riemann hypothesis.
| math_dandy wrote:
| Many heuristics in cryptanalysis rely on heuristics widely
| believed about the distribution of prime numbers. If RH is
| proved, some of these heuristics would become theorems.
|
| It would not, however, give practicing cryptanalysts new tools
| since belief in RH (and some even stronger conjectures!) are
| already baked in to the current toolbox.
| eigenvalue wrote:
| With these longstanding open problems, it's often not the
| result itself that is useful so much as the new techniques or
| the unification of previously unrelated branches of math that
| are invented/discovered to prove the result. Fermat's last
| theorem by itself didn't really lead to much new math, but the
| proof of it certainly did (and showed previously unknown
| relationships between modular forms and elliptic curves, and
| opened up new areas of research in number theory, algebraic
| geometry, and arithmetic geometry).
| wing-_-nuts wrote:
| Can I get a ELI!M[Not a mathematician]?
| nhatcher wrote:
| 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 wrote:
| 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
| seanhunter wrote:
| > 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
| dougbrochill wrote:
| For background, this video is a good overview of the Riemann
| Hypothesis: https://www.youtube.com/watch?v=d6c6uIyieoo
| chx wrote:
| Remember Indiana Jones and the Last Crusade? They are not in
| the chamber yet but they did disarm one of the traps in the
| temple.
| fishtockos wrote:
| More like Indy avoided drowning in the boat race scene (which
| I recall happening very early in the movie)
| QuesnayJr wrote:
| We have an approximate expression of how many prime numbers
| there are less than N, as N gets larger. If the Riemann
| hypothesis is true, then we know that the errors in this
| approximation are nice and small, which would allow us to prove
| many other approximate results. (There are many results of the
| form "If the Riemann hypothesis is true, then...")
| senderista wrote:
| _Prime Obsession_ is a good book-length intro to the RH (and to
| Riemann himself) that doesn't assume any math background.
| cgh wrote:
| Seconded, a great read and quite accessible. History chapters
| are interspersed with more mathematical ones. I would say
| having some basic math background is helpful but not totally
| necessary.
| cvoss wrote:
| 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 wrote:
| Aren't those sort of different things? I thought the whole
| point of provability was that it was distinct from truth.
| floxy wrote:
| 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 wrote:
| 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 wrote:
| 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.
| lisper wrote:
| > The word "finite" is doing some heavy lifting here - it
| is a second order concept.
|
| Sorry, you're going to have to explain that to me. The
| word "finite" has only one meaning AFAIK and on that
| definition it is definitely not the case that "every
| model of ZF contains a finite calculation for any non-
| trivial zero of the Riemann function." I don't even know
| what it means for ZF to "contain a calculation." The
| concept of calculation (at least the one that I'm
| familiar with) comes from computability theory, not
| number theory.
| lupire wrote:
| "finite" means what you think it means: a program that
| halts.
|
| Look at Peano Axioms:
|
| https://en.m.wikipedia.org/wiki/Peano_axioms
|
| The Induction axiom allows us to write finite proofs of
| otherwise infinite facts, like every positive integer has
| a predecessor.
|
| Without it, you'd be stuck with finitism (since only a
| finite number of numbers can be constructed in finite
| time, there is only a finite number of numbers.)
| lisper wrote:
| I think you're confusing PA with computability theory and
| Turing machines. "Program" is not a thing in PA.
| Tainnor wrote:
| PA can model Turing Machines and reason about them. I
| feel like you missed the important link between logic and
| computability theory.
| xrisk wrote:
| Didn't Godel show that in most useful logical systems there
| are true statements that cannot be proved?
| xavxav wrote:
| Indeed, the first incompleteness theorem tells us that
| any logical framework which can express Peano arithmetic
| must necessarily contain true (resp. false) facts for
| which no (resp. counter) proof can be given.
|
| Sometimes you can prove that no proof exists about a
| specific sentence (that's what his incompleteness proof
| does), and I think you could extend this technique to
| construct sentences where no proof exists of whether it
| has a proof, etc...
| Filligree wrote:
| The latter would be an axiom. A disproof would be a proof
| that there is no proof, so if you'd proven that no proof
| exists one way or the other then you've proven it can't
| be disproven _or_ proven.
|
| Which means you've hit a branch in mathematics. You can
| assume it to be either true or false, and you'll get new
| results based on that; both branches are equally valid.
| LegionMammal978 wrote:
| Constructively speaking, a disproof is a "proof that a
| statement leads to a contradiction". A "proof that there
| is no proof (assuming consistency)" can exist just fine,
| and that's exactly what the incompleteness theorem is,
| alongside a "proof that there is no disproof, i.e., that
| a contradiction cannot be derived from the statement
| (assuming consistency)".
| strbean wrote:
| > any logical framework which can express Peano
| arithmetic
|
| (with a finite list of axioms)
| jrvidal wrote:
| I think the precise pre-condition is that the theory
| should be recursive, which means either a finite list of
| axioms _or_ a computable check to determine whether a
| given formula is an axiom.
| lisper wrote:
| > the first incompleteness theorem tells us that any
| logical framework which can express Peano arithmetic must
| necessarily contain true (resp. false) facts for which no
| (resp. counter) proof can be given.
|
| Not quite. Any logical framework which can express Peano
| arithmetic must necessarily contain true facts for which
| no proof can be given _within PA_. The proof of Godel 's
| theorem itself is a (constructive!) proof of the truth of
| such a statement. It's just that Godel's proof cannot be
| rendered in PA, but even that is contingent on the
| assumption that PA consistent, which also cannot be
| proven within PA if PA is in fact consistent. In order to
| prove any of these things you need to transcend PA
| somehow.
| Tainnor wrote:
| > It's just that Godel's proof cannot be rendered in PA
|
| This is incorrect, the proof can be carried out in very
| weak subsystems of PA.
| zajio1am wrote:
| > must necessarily contain true (resp. false) facts for
| which no (resp. counter) proof can be given
|
| This formulation misses the important aspect that whether
| the statement is 'true' is not absolute property (outside
| logical truths). We can consider truthfulness of a
| statement in a specific structure or in a specific
| theory.
|
| E.g. a statement can be undecidable in Peano arithmetic
| (a theory) while true in natural numbers (a structure,
| model of Peano arithmetic), but that just means there is
| a different structure, different model of Peano
| arithmetic in which this statement is false.
| floxy wrote:
| https://en.wikipedia.org/wiki/Intuitionism
| hnfong wrote:
| No he showed _either_ there are true statements that
| cannot be proved, OR the system is inconsistent...
| lupire wrote:
| Probability is relative to an axiom systems.
|
| A more powerful system can prove statements that are true
| but not provable in a weaker system.
|
| For example "This sentence is not provable." (Godel's
| statement, written informally) is true, but not provable,
| in first order arithmetic logic.
|
| https://en.m.wikipedia.org/wiki/Diagonal_lemma
|
| In constructivist mathematics, proving that a statement is
| not false is not a proof that it is true. Non-
| constructivist mathematics, on the other hand, has the
| axiom, for all P: P or not-P, also stated as not-not-P
| implies P.
| alex_smart wrote:
| Not according to constructivists.
| zajio1am wrote:
| The whole point of provability is that it is purely syntactic
| process that could be verified in finite time. Ideally it
| would be the same as truth, but there are some caveats.
| riemann12 wrote:
| I've heard this argument:
|
| If RH is unprovable one way or another, then certainly no
| counterexample can exist to the RH otherwise you could find it
| and prove the RH to be false.
|
| Hence if RH is unprovable, it must be true. I suppose this uses
| logic outside the logical system that RH operates in.
| greenthrow wrote:
| 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 wrote:
| 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
|
| Your comment is rather condescending and kind of feels like a
| form of projection rather than a meaningful contribution.
| greenthrow wrote:
| 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 wrote:
| _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 wrote:
| Oh, I knew your username sounded familiar. Mister Smarty-Pansy-
| Cynical-McLoser back at it again!
| breck wrote:
| MathWorld on Riemann:
| https://mathworld.wolfram.com/RiemannZetaFunction.html
| blackle wrote:
| Relevant xkcd: https://xkcd.com/2595/
| GalaxyNova wrote:
| There's always a relevant xkcd isn't there
| nickcw wrote:
| 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=PLt5AfwLFPxWJdwkdjaK1o...
| nomilk wrote:
| TIL the Fields medal is only awarded to mathematicians under
| the age of 40.
|
| Source:
| https://www.youtube.com/watch?v=eupAXdWPvX8&list=PLt5AfwLFPx...
| adtac wrote:
| 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 wrote:
| The margin is too thin, in other words?
| manuelmoreale wrote:
| What a great reference! Well done.
| arrowsmith wrote:
| I don't get it.
| seanhunter wrote:
| 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
| manuelmoreale wrote:
| Thank you for explaining it way better than I could have
| done
| lupire wrote:
| The 2022 Fields medal winners were born in 1983, 1984,
| 1985, and 1987 (Maynard).
| chongli wrote:
| I went over this table of Fields Medal winners [1] and the
| 4n+2 people have won 28% of the awards. However, 33% of the
| awards were won by 4n+3 people but only 17% were won by
| 4n+0 people.
|
| So your hypothesis does seem to bear out: people born on
| 4n+0 years are at a significant disadvantage for winning a
| Fields medal.
|
| [1] https://stats.areppim.com/listes/list_fieldsxmedal.htm
| hyperbovine wrote:
| Error bars please.
| Sniffnoy wrote:
| Yes, I don't think the Fields medal was intended as the
| "Nobel prize of mathematics", but since it was the biggest
| award that existed it got promoted as such despite its
| inequivalence. More recently there's the Abel prize, which
| tries to be a more direct Nobel prize analogue, but of course
| the Fields medal has a multi-decade head start in terms of
| promotion...
| aleph_minus_one wrote:
| Nobel Prize (and Abel Prize and Wolf Prize) are for the
| lifetime achievement, i.e. the scientific contributions
| that lead to the prize might be from decades ago.
| ks2048 wrote:
| Maynard's videos on Numberphile are great. Like Tao, he seems
| to be able to explain things clearly to non-experts - I think
| another sign of greatness.
| samscully wrote:
| 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=PLbaA3qJlbE...
| amirhirsch wrote:
| 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
|
| 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 k _pi 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
| atonalfreerider wrote:
| 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-visualizatio...
| amirhirsch wrote:
| 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 wrote:
| Any good easily digestible sources on this topic?
| panstromek wrote:
| Probably the 3blue1brown video:
| https://youtu.be/sD0NjbwqlYw?si=T86Um-4Bj2WBIf3n
| arrowsmith wrote:
| That video was really great - thanks for sharing.
| WoahNoun wrote:
| Prime Obsession by John Derbyshire
| md224 wrote:
| Aw man, yours is way cooler than mine: https://matt-
| diamond.com/zeta.html
|
| Still, it's funny to see how many people have attempted this!
| It's a fun programming exercise with a nice result.
| nyc111 wrote:
| What is the actual formula you are using to plot the graph?
| brcmthrowaway wrote:
| 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 wrote:
| 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 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 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 wrote:
| [1] neither
|
| [2] neither
|
| :) but I'm glad there are people out there that do!
| billforsternz wrote:
| 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 wrote:
| "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. 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 (a slight improvement of
| 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 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.
| billforsternz wrote:
| Thank you! I periodically remember this and for years I've
| been meaning to try to find out for sure whether I had
| somehow just misunderstood completely. Very pleased to know
| for sure that I had and that the Riemann Hypothesis remains a
| genuine mystery.
| Havoc wrote:
| No idea what any of that means but they're clearly stoked about
| it so happy for them
| nsoonhui wrote:
| 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)
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