A breakthrough towards the Riemann hypothesis(mathstodon.xyz)
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A breakthrough towards the Riemann hypothesis
https://mathstodon.xyz/@tao/112557248794707738
177 comments
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.
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.
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...
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...
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.
Still, it's funny to see how many people have attempted this! It's a fun programming exercise with a nice result.
What is the actual formula you are using to plot the graph?
For the longest time I thought the zeta curve was some kind of sophisticated equation, but it is astonishingly simple. The "magic" of the zeta zeros only happens because of the 1/2 term in the exponent of the equation below. Any change with this fraction, and the zeros do not converge.
You start with a line segment. You then draw another line segment that starts at the end of the previous line segment, and whose length is shorter than the previous segment. The length of any segment is (1/n)^(1/2) where n is the number of the segment. These segments approach a limit (think of Zeno's paradox).
You start with a line segment. You then draw another line segment that starts at the end of the previous line segment, and whose length is shorter than the previous segment. The length of any segment is (1/n)^(1/2) where n is the number of the segment. These segments approach a limit (think of Zeno's paradox).
_ <- segment 3
segment 2 -> /
/ <- bend alpha between segments
|
segment 1 -> |
|
|
Any good easily digestible sources on this topic?
Probably the 3blue1brown video: https://youtu.be/sD0NjbwqlYw?si=T86Um-4Bj2WBIf3n
That video was really great - thanks for sharing.
Prime Obsession by John Derbyshire
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...
https://www.youtube.com/playlist?list=PLt5AfwLFPxWJdwkdjaK1o...
TIL the Fields medal is only awarded to mathematicians under the age of 40.
Source: https://www.youtube.com/watch?v=eupAXdWPvX8&list=PLt5AfwLFPx...
Source: https://www.youtube.com/watch?v=eupAXdWPvX8&list=PLt5AfwLFPx...
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.
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.
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
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
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Error bars please.
The margin is too thin, in other words?
What a great reference! Well done.
I don't get it.
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
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
Thank you for explaining it way better than I could have done
The 2022 Fields medal winners were born in 1983, 1984, 1985, and 1987 (Maynard).
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...
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.
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.
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...
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...
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."
"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."
It's perfectly common for a mathematician to successfully use a technique where another top mathematician tried and failed.
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.
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.
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.
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)
[0]: https://en.wikipedia.org/wiki/Larry_Guth
[1]: https://en.wikipedia.org/wiki/James_Maynard_(mathematician)
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.
Maynard won a Fields medal, Tao is of course in the elite, but so is JM.
Maynard is a Fields Medallist, so is also one of the strongest mathematicians around.
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.
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.
Tao and Maynard are both academics…
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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...
[1] https://www.sciencenews.org/article/why-we-care-riemann-hypo...
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').
> (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').
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.
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.
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.
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.
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:
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:
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)
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
2 1.5 x 10^45
3 1.3 x 10^51
4 1.1 x 10^56
5 2.1 x 10^60
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. ;)
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. ;)
Of course, if you end up with an incorrect result due to assuming the Riemann hypothesis, you'll get a fields medal anyway...
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
For 64 bit primes they would need to do around 48 tests per prime to get down to a similar error rate.
For
Then why not just assume it's true and if something breaks, hey, you might just have disproved it?
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.
Then it seems it wouldn't be better to use the deterministic test ever. Just use the randomised test and accept the failure rate.
What if an airliner or a spacecraft or a self- driving car or a surgical robot breaks?
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.
They already do. Was this supposed to be an enlightening question?
as nomilk said above, for any practical real-world application you could likely just assume the reimann hypothesis was true and proceed accordingly
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.
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.
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)
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.
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.
Aren't those sort of different things? I thought the whole point of provability was that it was distinct from truth.
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.
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."
Didn’t Gödel show that in most useful logical systems there are true statements that cannot be proved?
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...
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...
> 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.
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.
> 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.
This is incorrect, the proof can be carried out in very weak subsystems of PA.
> 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.
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.
Discussing Gödel in terms of "truth" immediately brings about deep philosophical discussions about what that actually means in the context of mathematics. If the "true" nature of the (standard model of) arithmetic is unknowable in full, does it even exist?
I like to be pragmatically classical in my mathematical outlook (I don't worry about LEM), but when we come to foundations, I find that we need to be a little bit more careful.
Gödel's original proof (or rather, the slightly modified Gödel-Rosser proof) avoids notions of absolute truth - it says that for any consistent model of arithmetic, there must exist undecidable sentences. These are ultimately purely syntactical considerations.
(I'm not claiming that there is no relation to "truth" at all in this statement, just that things become less precise and more philosophical once that's how we're going to frame things)
I like to be pragmatically classical in my mathematical outlook (I don't worry about LEM), but when we come to foundations, I find that we need to be a little bit more careful.
Gödel's original proof (or rather, the slightly modified Gödel-Rosser proof) avoids notions of absolute truth - it says that for any consistent model of arithmetic, there must exist undecidable sentences. These are ultimately purely syntactical considerations.
(I'm not claiming that there is no relation to "truth" at all in this statement, just that things become less precise and more philosophical once that's how we're going to frame things)
These deep philosophical discussions are about absolute "truth" in general, but in logic, the truthness of formula in specific mathematical structure is well-defined through structural induction over the formula, although with non-finite, non-computable steps. Yes, from strict-finitistic point of view even the concept of truth in the standard model of arithmetic could be problematic.
I'm personally not a constructivist or even finitist (mostly for pragmatic reasons), but discussions about Gödel get sidetracked by musings about either of these philosophies so often that I feel it's important to point out that there is an understanding of Gödel's theorems that can be understood and discussed regardless of one's commitment to a particular philosophy of mathematics (well, at least assuming some "reasonable" base).
If you are a finitist, you could interpret Gödel's theorem to mean that infinite systems such as the natural numbers don't actually exist, because they can never be understood in full (I'm not a finitist, so maybe I'm wrong in this conclusion). If you're a classical platonist, they would mean that the "natural numbers" do exist in some platonic realm, but will never fully be captured by human mathematics. If you're a formalist like me, maybe you'd say that it's very useful to pretend that the natural numbers exist, but that strictly speaking we don't really fully understand what they are or should be (but it turns out not to matter all that much).
Either way, a complete theory of the natural numbers doesn't exist.
If you are a finitist, you could interpret Gödel's theorem to mean that infinite systems such as the natural numbers don't actually exist, because they can never be understood in full (I'm not a finitist, so maybe I'm wrong in this conclusion). If you're a classical platonist, they would mean that the "natural numbers" do exist in some platonic realm, but will never fully be captured by human mathematics. If you're a formalist like me, maybe you'd say that it's very useful to pretend that the natural numbers exist, but that strictly speaking we don't really fully understand what they are or should be (but it turns out not to matter all that much).
Either way, a complete theory of the natural numbers doesn't exist.
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.
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.
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)".
No he showed either there are true statements that cannot be proved, OR the system is inconsistent...
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.
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.
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.
> 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.
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.
How much explanation do you want?
Any consistent first order set of axioms that has a model, has multiple models. In the case of the Peano Axioms, there is one and only one model that we're interested in - the finite integers. All the others we call nonstandard.
But there is absolutely no statement in first order logic that can explain what we mean by finite, and why THIS model is what we want, and THAT model should be considered nonstandard.
However there is something called second order logic. As https://www.lesswrong.com/posts/3FoMuCLqZggTxoC3S/logical-pi... correctly says, in second order logic we really can say that something is finite, and mean it. But as https://www.lesswrong.com/posts/SWn4rqdycu83ikfBa/second-ord... explains, anyone with Constructivist tendencies is going to look at second order logic's definition of finite and call it wordy hogwash.
Regardless, we can look at things this way. Given any two models of the same set of axioms, we can try to set up a correspondence between the models. Each may contain some things not in the other. For example a model of PA + (PA is inconsistent) will contain a proof of inconsistency that is not in a standard model of PA.
The standard model of PA is simply this, it is a model of PA which is contained in every other model of PA. And so, whether or not two models agree on the Riemann Hypothesis, they will all agree on what the billionth zero is. And the trillionth. And so on for every finite number. Why? Because those are all in the standard model. And everything in the standard model is included in every other.
Any consistent first order set of axioms that has a model, has multiple models. In the case of the Peano Axioms, there is one and only one model that we're interested in - the finite integers. All the others we call nonstandard.
But there is absolutely no statement in first order logic that can explain what we mean by finite, and why THIS model is what we want, and THAT model should be considered nonstandard.
However there is something called second order logic. As https://www.lesswrong.com/posts/3FoMuCLqZggTxoC3S/logical-pi... correctly says, in second order logic we really can say that something is finite, and mean it. But as https://www.lesswrong.com/posts/SWn4rqdycu83ikfBa/second-ord... explains, anyone with Constructivist tendencies is going to look at second order logic's definition of finite and call it wordy hogwash.
Regardless, we can look at things this way. Given any two models of the same set of axioms, we can try to set up a correspondence between the models. Each may contain some things not in the other. For example a model of PA + (PA is inconsistent) will contain a proof of inconsistency that is not in a standard model of PA.
The standard model of PA is simply this, it is a model of PA which is contained in every other model of PA. And so, whether or not two models agree on the Riemann Hypothesis, they will all agree on what the billionth zero is. And the trillionth. And so on for every finite number. Why? Because those are all in the standard model. And everything in the standard model is included in every other.
Thanks, that was very helpful. One question though:
> there is absolutely no statement in first order logic that can explain what we mean by finite
That seems implausible. All natural numbers are finite. I can easily express that in F.O.L.
> there is absolutely no statement in first order logic that can explain what we mean by finite
That seems implausible. All natural numbers are finite. I can easily express that in F.O.L.
> That seems implausible. All natural numbers are finite. I can easily express that in F.O.L.
Just for the sake of clear terminology, I'll take "finite natural number" to mean that it's either 0 or formed by applying the successor function to zero finitely many times.
Here's an easy proof for why you can't prove that every number is finite in PA:
Take the theory PA' = PA + the infinitely many sentences "c > 0", "c > S 0", "c > S (S 0)", etc. (with a new constant c)
We can probably agree that the standard natural numbers aren't a model of PA' because there is no natural number greater than every other natural number.
However, the natural numbers are clearly a model of any finite subset of that theory. And here's where we use the compactness theorem (which is easily proven from completeness): https://en.m.wikipedia.org/wiki/Compactness_theorem
By compactness, PA' has a model. But that model can't be the standard model of arithmetic. Yet it is also a model of PA. Thus we have a model of PA in which infinite numbers exist - so we can't prove that all natural numbers are finite from PA.
To rule out such models - and therefore prove that every natural number is finite - you need the second-order induction axiom.
Just for the sake of clear terminology, I'll take "finite natural number" to mean that it's either 0 or formed by applying the successor function to zero finitely many times.
Here's an easy proof for why you can't prove that every number is finite in PA:
Take the theory PA' = PA + the infinitely many sentences "c > 0", "c > S 0", "c > S (S 0)", etc. (with a new constant c)
We can probably agree that the standard natural numbers aren't a model of PA' because there is no natural number greater than every other natural number.
However, the natural numbers are clearly a model of any finite subset of that theory. And here's where we use the compactness theorem (which is easily proven from completeness): https://en.m.wikipedia.org/wiki/Compactness_theorem
By compactness, PA' has a model. But that model can't be the standard model of arithmetic. Yet it is also a model of PA. Thus we have a model of PA in which infinite numbers exist - so we can't prove that all natural numbers are finite from PA.
To rule out such models - and therefore prove that every natural number is finite - you need the second-order induction axiom.
> To rule out such models - and therefore prove that every natural number is finite - you need the second-order induction axiom.
Sorry, I don't see why second-order induction is required. AFAICT all you need is a first-order induction axiom instantiated on the predicated FINITE.
Sorry, I don't see why second-order induction is required. AFAICT all you need is a first-order induction axiom instantiated on the predicated FINITE.
I don't know why I wrote an entire proof outlining why first-order PA cannot rule out infinite numbers, if you just ignore it and insist on your misconception. The set of finite numbers is undefinable in the language of first-order PA, which is why first-order induction isn't enough.
Read a textbook, these are well-known facts. Here's a free one that is quite good, chapter 3 is relevant: https://milneopentextbooks.org/a-friendly-introduction-to-ma...
Read a textbook, these are well-known facts. Here's a free one that is quite good, chapter 3 is relevant: https://milneopentextbooks.org/a-friendly-introduction-to-ma...
Well, you can define something meaning "finite" in first order logic. It just won't mean what we want it to mean.
Take your example. "All natural numbers are finite." How do we define the natural numbers in first order logic? Let's use PA. Now take a nonstandard model of PA in which you can prove that PA is inconsistent. You will have a "proof" in that model that PA is inconsistent. That "proof" has a length. In that model, that "proof" will be of "finite" length.
There is no contradiction here...but only because that "proof" is (from our point of view) infinitely long. So the proof can go wrong even though it goes wrong at no step.
From my point of view, first order logic captures the idea of "finite" as well (or poorly) as classical mathematics captures the idea of "exists". And second order logic requires accepting the truth of unprovable things. But if you're willing to accept the truth of unprovable things, and the existence of non-constructible ones, we get the standard mathematics that mathematicians use without questioning too hard.
Take your example. "All natural numbers are finite." How do we define the natural numbers in first order logic? Let's use PA. Now take a nonstandard model of PA in which you can prove that PA is inconsistent. You will have a "proof" in that model that PA is inconsistent. That "proof" has a length. In that model, that "proof" will be of "finite" length.
There is no contradiction here...but only because that "proof" is (from our point of view) infinitely long. So the proof can go wrong even though it goes wrong at no step.
From my point of view, first order logic captures the idea of "finite" as well (or poorly) as classical mathematics captures the idea of "exists". And second order logic requires accepting the truth of unprovable things. But if you're willing to accept the truth of unprovable things, and the existence of non-constructible ones, we get the standard mathematics that mathematicians use without questioning too hard.
> Now take a nonstandard model of PA in which you can prove that PA is inconsistent.
Sorry, I lost you there. You don't prove things in models, you prove things in systems. Models are semantic things, proofs are syntactic things, so a "proof in a model" sounds like a category error to me.
Sorry, I lost you there. You don't prove things in models, you prove things in systems. Models are semantic things, proofs are syntactic things, so a "proof in a model" sounds like a category error to me.
Proofs are natural numbers via Gödel's encoding, or any other encoding scheme that you'd like (e.g. ASCII). The sentence "PA is inconsistent" can be stated in the language of PA and it means "there is a number that encodes a valid proof from the axioms of PA to a contradiction (e.g. 0 ≠ 0)". This sentence is either true in all models of PA (in which case PA would be inconsistent, which we don't think it is), or it is true in some models and false in others - it can't be false in all models because otherwise, by completeness, there'd be a proof of PA's consistency in PA which is what the second incompleteness theorem rules out.
The reason why this is not a category error is because of the self-referentiality of strong enough systems of logic.
The reason why this is not a category error is because of the self-referentiality of strong enough systems of logic.
I guess I calibrated your understanding of formal logic to the wrong place. Let me try again.
PA is a set of axioms satisfied by the natural numbers. As Gödel showed, all of first order logic can be encoded into the natural numbers. And questions like, "This is a proof of X" can be viewed as arithmetic statements. (We can actually go further, and encode all of computation into the natural numbers, and reason about computation using nothing more than PA.)
Gödel famously went further. Suppose that X is an axiom system. Then Gödel's famous incompleteness theorem says:
It is very important here to note. If we actually have a proof from X of Con(X), then Gödel actually gives us a proof from X of not Con(X).
But, thanks to the Gödel numbering, (X => Con(X)) can also be interpreted as a statement about the natural numbers. As such, (X => Con(X)) is a potential axiom in first order logic. Of course all proofs that "really are proofs" correspond to actual finite numbers. So if there "isn't really" a proof of Con(X) from X, then any model of X + (X => Con(X)) will have the Gödel number of that proof not be a finite number. But that's OK because first order logic can't rule out nonstandard models. And nonstandard models of PA include infinite integers. So any model of X + (X => Con(X)) will have a "proof" of X + (X => Con(X)). We can decode as much of that "proof" as we like. But we can't decode the whole thing, and it "isn't really" a proof after all.
Of course if we have that then Gödel's construction gives us a Gödel number for the proof of not Con(X).
This is an absolutely general line of reasoning. Take any consistent set of first order axioms X which imply PA. If it is consistent, we don't create a contradiction by adding the axiom that there exists a Gödel number representing (X => Con(X)). Because this new set of axioms is consistent, there must exist (for some definition of existence) a model for it. That model for it will include a model of PA. That model of PA will include a Gödel number for a proof that isn't really a proof, meaning that it can't really a be finite integer. Therefore it must be an infinite integer. And therefore the set of first order axioms X cannot distinguish between what we want finite to mean, and something that is clearly not finite.
Second order logic gets around this problem. But it does so by accepting that things can be true without there being any proof that it is true. This is one of the potential sticking points for someone with Constructivist tendencies.
PA is a set of axioms satisfied by the natural numbers. As Gödel showed, all of first order logic can be encoded into the natural numbers. And questions like, "This is a proof of X" can be viewed as arithmetic statements. (We can actually go further, and encode all of computation into the natural numbers, and reason about computation using nothing more than PA.)
Gödel famously went further. Suppose that X is an axiom system. Then Gödel's famous incompleteness theorem says:
((X => PA) + (X => Con(X))) => not Con(X)
It is very important here to note. If we actually have a proof from X of Con(X), then Gödel actually gives us a proof from X of not Con(X).
But, thanks to the Gödel numbering, (X => Con(X)) can also be interpreted as a statement about the natural numbers. As such, (X => Con(X)) is a potential axiom in first order logic. Of course all proofs that "really are proofs" correspond to actual finite numbers. So if there "isn't really" a proof of Con(X) from X, then any model of X + (X => Con(X)) will have the Gödel number of that proof not be a finite number. But that's OK because first order logic can't rule out nonstandard models. And nonstandard models of PA include infinite integers. So any model of X + (X => Con(X)) will have a "proof" of X + (X => Con(X)). We can decode as much of that "proof" as we like. But we can't decode the whole thing, and it "isn't really" a proof after all.
Of course if we have that then Gödel's construction gives us a Gödel number for the proof of not Con(X).
This is an absolutely general line of reasoning. Take any consistent set of first order axioms X which imply PA. If it is consistent, we don't create a contradiction by adding the axiom that there exists a Gödel number representing (X => Con(X)). Because this new set of axioms is consistent, there must exist (for some definition of existence) a model for it. That model for it will include a model of PA. That model of PA will include a Gödel number for a proof that isn't really a proof, meaning that it can't really a be finite integer. Therefore it must be an infinite integer. And therefore the set of first order axioms X cannot distinguish between what we want finite to mean, and something that is clearly not finite.
Second order logic gets around this problem. But it does so by accepting that things can be true without there being any proof that it is true. This is one of the potential sticking points for someone with Constructivist tendencies.
Thanks for taking the time to explain this. I thought I understood this stuff, but apparently I'm still missing something. I obviously need to ponder it some more, because I still don't understand why second-order induction succeeds where first-order induction fails. We are only dealing with one predicate here, FINITE. So why should it make a difference if induction on FINITE is a first-order instantiation of an axiom schema, or an instantiation of a second-order axiom? Proving that all natural numbers are finite still boils down to induction on FINITE, no?
"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.)
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.)
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.
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.
Not according to constructivists.
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.
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.
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.
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.
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.
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.
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?
Just this one?
Oh, I knew your username sounded familiar. Mister Smarty-Pansy-Cynical-McLoser back at it again!
Can I get a ELI!M[Not a mathematician]?
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.
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.
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
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
> 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
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
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.
More like Indy avoided drowning in the boat race scene (which I recall happening very early in the movie)
For background, this video is a good overview of the Riemann Hypothesis: https://www.youtube.com/watch?v=d6c6uIyieoo
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...")
_Prime Obsession_ is a good book-length intro to the RH (and to Riemann himself) that doesn't assume any math background.
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.
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.
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.
"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.
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.
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.
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!
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!
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
His masterclass is substantially better than the average masterclass, particularly for somebody who isn't already an amateur or better mathematician.
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/
His book Analysis I, and potentially also his linear algebra lecture come to mind.
What are the practical implications of a conclusion of the hypothesis, in either direction?
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.
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."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.
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.
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.
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.
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.
"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.
I trust it's as wrong now as it always has been.
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.
But that’s a lot of what ifs away.
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.
Not RH being true, but that the proof itself would require discovering and proving something of high interest as an intermediate step.
Perhaps, but it’s not clear that would be cryptographically relevant still.
Testing primality doesn’t break encryption; we already test for primarily on a daily basis very efficiently
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.
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.
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.
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).
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.
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.
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MathWorld on Riemann: https://mathworld.wolfram.com/RiemannZetaFunction.html
This improves modeling of long-tail distributions? How far off am I here?
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.)
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.)
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
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
My bad. Thanks for the correction.
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Can you be more specific?
This is a result about a specific approximately known distribution.
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.
I'm wondering if this result indicates we have a new method to eke out important signals from noise that otherwise get smoothed out.
Is there any chance they made this talk available online?
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
It makes no sense if the rest of the universe relies upon simple rules
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.
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.
[1] neither
[2] neither
:) but I'm glad there are people out there that do!
[2] neither
:) but I'm glad there are people out there that do!
No idea what any of that means but they’re clearly stoked about it so happy for them
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.
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.
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.
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.
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.
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.
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.
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.
> 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.
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.
> 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!
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!
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...)
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...)
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.
Your other points are similar.
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.
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.
Time will tell.
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.
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.
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Yeah, random people on HN are much better judges of importance of mathematical results.
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!
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.
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.
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.
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.
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.
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.
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.
Change that to CPU performance in 2023 and you got a breakthrough as well.
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pheatherlite(4)
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