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 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
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.
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).
Finally, you bend each segment by an angle alpha. Technically this angle is in imaginary space, but the visual in Cartesian space just looks like a spiral, where each bend adds an angle, like a bull whip, so that the whole curve spirals back around (after creating other, mesmerizing sub-spirals). Amazingly, this curve always intersects zero (per Riemann Hypothesis). As I mentioned in my other comment, it's very useful to see this curve in 3D space.
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:
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.
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.
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.
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.
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!
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."
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.
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.
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.
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').
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.
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.
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
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. ;)
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.
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.
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.
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.
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.
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."
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...
> 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.
> 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.
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)
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.
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.
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)".
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.
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.
> 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.
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.
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.
> 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.
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.
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.
> 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.
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.
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:
((X => PA) + (X => Con(X))) => not Con(X)
Let's take that apart. By (X => PA) I mean that from X we can find the existence of something that satisfies the Peano axioms. By (X => Con(X)) I mean that there is a proof from X that X is consistent. And from those facts, Gödel can construct a proof from X that X is not consistent.
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?
So the problem with logic and in particular model theory is that you always have to assume some sort of logic for the world where your models live in (a meta-logic) - otherwise you can't talk about anything.
If you take e.g. ZFC for granted in the world of your models, then you can talk about "the" natural numbers, where every number is finite, as a model of PA. And you can also define non-standard models of PA which include infinite numbers. If you use some other foundation, you can probably also define these models in some way. Now whether these constructions mean anything to you, or capture your intuition about "finiteness" is another thing entirely...
The problem is that from within the theory you cannot distinguish between these different models. There's no way of defining a predicate "finite" in first-order PA that agrees with the notion of "finite" that you've established in your meta-logic.
And in general, it's worse than that. As long as your first-order theory allows for any infinite model, there exist models of that theory of any infinite cardinality (this is the Löwenheim-Skolem theorem). So, in general, first-order logic cannot pin down structures uniquely. (Again, this is assuming ZFC in the world of our models.)
This btw also leads to the "Skolem paradox" which states that there are countable models of (first-order) ZFC, even though ZFC says that there exist uncountable sets. But this is only a "paradox" until you realise that "countable" and "uncountable" don't mean the same thing within the model as they do outside of the model.
> There's no way of defining a predicate "finite" in first-order PA that agrees with the notion of "finite" that you've established in your meta-logic.
Why doesn't this work?
Define the predecessor function P such that P(S(n)) = n for all n (and maybe S(P(n)) = n for all n != 0).
FINITE(n) := (n=0) OR FINITE(P(n))
i.e. a number is finite if it is zero or if it predecessor is finite.
Why doesn't that work?
BTW, as an aside, since you seem to know your stuff I thought I'd ask: I noticed as part of a separate discussion over on /r/math that AFAICT there is nothing in PA that distinguishes successor and predecessor until you get to the definition of multiplication. Is that right? Is it noteworthy?
> I noticed as part of a separate discussion over on /r/math that AFAICT there is nothing in PA that distinguishes successor and predecessor until you get to the definition of multiplication. Is that right? Is it noteworthy?
I'm not entirely sure what you mean by this. "PA without multiplication" is basically Presburger Arithmetic (https://en.m.wikipedia.org/wiki/Presburger_arithmetic). One of its axioms (as in PA) is that 0 has no predecessor - but clearly every number, including 0, has a successor. So that's a way of distinguishing predecessor and successor functions.
I believe that when you look at nonstandard models (which basically look like N + any number of copies of Z), the order isn't fixed in those copies of Z (unlike N), so you could swap predecessor and succesor within those copies, but I haven't really looked into that all that much.
Yeah, that's not quite what I meant. What I was trying to get at was more analogous to the ambiguity of i and -i. Suppose I wanted to create a system of math whose intended model was the negative naturals rather than the positive naturals. Let's call this new system NPA. NPA has the exact same axioms as PA except with predecessor instead of successor as the core axiomatic construct. So zero has a predecessor, but no successor. Everything works as before, except with the letter P instead of the letter S. Addition still works, you're just adding negative numbers and getting negative numbers as the result, which is what you would expect. It doesn't break down until you get to multiplication, where, if you follow the same definition as standard PA you get the "wrong" result because you want the product of two negative numbers to be a positive number, and there are no positive numbers in NPA. So AFAICT multiplication is required to distinguish PA from NPA. Which raises an interesting question: is there a non-trivial operation you can define to distinguish i from -i the way that multiplication allows you to distinguish 1 from -1?
In every model of a theory, you can just consistently relabel elements in order to get another model. But that doesn't really tell you anything because that model is just isomorphic to the old one (that term has a precise meaning, but I think the "relabeling" intuition captures it).
So your NPA is just a consistent relabeling of elements n to elements -n - it's fully isomorphic to PA. There's nothing particularly interesting about it. Notice that "the product of two negative numbers is positive" doesn't help because in your alternative model, "positive" (or rather, "nonnegative") just means something different (or more precisely, it doesn't even make sense to talk about "nonnegative numbers" because that's all of them).
It only becomes interesting when you have different non-isomorphic models of a theory - that then means that there are (distinguishable) properties of the model that the theory can't talk about.
In the case of the complex numbers, if you just exchange i and -i, I believe that you get a field which is fully isomorphic to C.
> In the case of the complex numbers, if you just exchange i and -i, I believe that you get a field which is fully isomorphic to C.
To add to this and expand a little, there are only two isomorphisms between C and itself which keep all the real numbers fixed: the identity and the conjugate map (which sends a+ib to a-ib, and in particular i to -i).
If you don't fix the real numbers there are other such isomorphisms but I believe you can only construct them via the axiom of choice (so you can't really "write them down").
In the case of the real numbers there's really only one such isomorphism, the identity.
Right. So in those terms, NPA is isomorphic to PA -- until you introduce multiplication, at which point the isomorphism goes away becauase S(0)S(0) is S(0) but P(0)P(0) is not P(0). That seems like an interesting fact to me, and feels like it might be a clue pointing to some deeper truth.
If you start with NPA it is identical as PA. You can apply all the same logic and all the first theorems. The problem with multiplication is not that the proofs fall apart, it is that multiplication does not mean what you want it to mean.
When you go to the complex numbers, you have a similar situation. The only thing that distinguishes i from -i is what you want the principal branch of sqrt to be. Otherwise nothing would change if you flip them.
> The problem with multiplication is not that the proofs fall apart, it is that multiplication does not mean what you want it to mean.
Well, yeah, that's kind of the point: the positiveness of the naturals is not just a convention, it is actually a logical consequence of something in the axioms. Specifically, it is a logical consequence of the definition of multiplication. I find that interesting.
That ties to one of the standard axioms of the real numbers in classical mathematics.
There exists a set P which is closed under addition and multiplication, such that for every real number x, one of these three things is true: x is in P, -x is in P, x = 0.
P is, of course, the positive numbers. This axiom makes the reals into an ordered field. Add in completeness, and the reals are unique (up to choice of set theory).
You can have an infinite chain of predecessors of non-standard numbers, each claiming being FINITE(). That satisfies your definition. This could be easily demonstrated in Robinson arithmetic, which has simple, understandable non-standard models.
Using ZFC, Robinson can take a standard model of the reals and construct a nonstandard model, and a function from the standard model to the nonstandard model. Every possible statement of first order logic that can be made about the standard model is true in the nonstandard model. Likewise every possible statement of first order logic that can be made in the nonstandard model only involving things that came from the standard model are also true of the standard model. This is his "transfer principle". However the nonstandard model comes with things like infinitesmals, and infinite numbers. So in the nonstandard world we can really do things like define a derivative as dy/dx, or an integral as an infinite sum. Then recover the classical definitions through the transfer principle.
See https://terrytao.wordpress.com/2007/06/25/ultrafilters-nonst... for an explanation of how this actually works in some detail. But if you get your head around it, you'll see exactly why first order logic cannot possibly capture the idea of being a standard model. Because, in first order logic, the nonstandard model really is indistinguishable. And so finite cannot be expressed in first order logic.
> You can have an infinite chain of predecessors of non-standard numbers, each claiming being FINITE().
Ah. That makes sense. Thanks.
So surely someone has come up with a system that captures the intuitive spirit of mathematical induction, i.e. that the chain has to actually be rooted at 0 rather than have an infinite chain of predecessors or be a loop? How hard could that be?
It seems like it should be straightforward. But it's mathematically impossible.
Similarly, it seemed to David Hilbert like it should be possible to have a system of formal logic that all could accept, which would prove itself to be consistent. But that also proved to be impossible. And the logic which lead to that being proved impossible is deeply connected to the first line of reasoning that I gave you for why this is also impossible.
And in the end what they all come down to is some kind of self-reference. In this case, to have a finite statement define what it means to be finite, we must already know what finite means. Second order logic sort of assumes that we can simply know what finite means. First order logic doesn't allow you to wave your hands like that. You can't cheat in any way. And so you can't find a way to do it.
> Similarly, it seemed to David Hilbert like it should be possible to have a system of formal logic that all could accept, which would prove itself to be consistent.
Minor nitpick, but I don't think this fully captures the essence of Hilbert's program. If we had a system that could prove itself consistent, that would mean nothing, because the system could be inconsistent, therefore unsound and so its consistency "proof" wouldn't mean anything.
Rather, the goal was to start from an arguably "uncontroversial" base of mathematics (something like primitive recursive arithmetic, which intuitionists might also accept) and use that to bootstrap "infinitary mathematics" of the kind the formalists such as Hilbert were arguing for. The hope was that one could prove the consistency of such more abstract mathematics (PA or ZFC) from the more concrete foundations.
Unfortunately, Gödel's theorems completely demolish that idea. It's not just that a (sufficiently strong) theory can't prove itself consistent, it also can't be proven consistent by a strictly weaker theory such as PRA (because if such a proof existed in the weaker system, it would of course also exist in original system).
> Second order logic sort of assumes that we can simply know what finite means. First order logic doesn't allow you to wave your hands like that.
OK, this I do not understand. How does 2ndOL "sort of assume that we can simply know what finite means"? That seems like an extraordinarily imprecise characterization of 2ndOL. 2ndOL is a formal system just like 1stOL. Neither one lets you "wave your hands" about anything AFAICT.
I view the "for all properties" bit as handwaving. See particularly section 6 of that article, which at one point gives:
The semantics of second-order logic depending on these hard questions of set theory puts second-order logic into the same “basket” with set theory. Criticism of set theory becomes criticism of second-order logic and vice versa.
My Constructivist tendencies are due to my discomfort with accepting the validity of set-theoretic constructions with decisions based on unverifiable truths. Accepting second order logic requires accepting quantifying over similar collections of unverifiable claims. I'm going to be no more sympathetic.
In any model of PA, all natural numbers will be finite by this definition. Even if we, from outside of the model, know that many of them aren't.
You've been presented with two different ways to prove this, and a standard textbook. Perhaps read the textbook?
On your aside, in PA there is one element that is not the successor of any other. And it likewise has no predecessor. (It is a matter of convention whether you call that special number 0 or 1.) That special number is always the base case of all induction proofs.
You list things that you want to be true of it. The Peano Axioms are examples. But no matter how long your list gets, it can always get longer. For example we can ask whether a particular polynomial has integer roots. See https://en.wikipedia.org/wiki/List_of_statements_independent.... (It doesn't, but you can't prove that in ZFC.)
Even so, there is always an, "...and so on." You never finish, because finite does not actually have a finite definition.
Second order logic gets around it because you can say something that simultaneously implies all the things that you want to say about "finite" in one go. Such as, "All possible functions from itself to itself that are onto are also one-to-one." A first order model need not capture the full intent of this statement, because it does not contain all possible functions. As we can see when we look at a model from within other models.
But when I look at what the second order idea of "all possible functions" actually means, I think we're just begging the question of what it means to be finite.
The way that I think about it is that some things are finite. Some things are infinite. And some things on the boundary are not yet determined. But then again, I do have Constructivist sympathies. And this is not a point of view that most mathematicians are very sympathetic towards.
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.)
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.
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:
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.
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.
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.
> 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.
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...")
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.
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).
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.
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
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/
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."
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.
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.
"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.
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.
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.
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.
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.)
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.
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.
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.
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.
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.
> 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
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.
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...)
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.
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.
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.
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.
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.
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.
I think giving any more details than his three short posts would delve too far into the specific arguments of the paper to be valuable for a wide enough audience, subtracting those that are going to be looking at the pre-print anyhow.
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