 Thank you very much for this opportunity to speak in your web seminar. So yeah, there's a lot here, a lot of topics, and I put together a little graph to show the connection between these various topics, both, how to say, existing connections and some new ideas, some new connections. So the main object was to study gaps between primes, in particular the large gaps, and there are connections with something called the interval sieve, probabilistic models, which are somewhat familiar to a lot of people, the Hardy-Littlewood K-tuples conjecture, and the existence of z equals zero. Now I've drawn a bunch of arrows here showing the connections, and for some of them, I've written names of people who have done the work to make these connections. And then there's a brand new one, which I threw in just recently because Andrew Granville shared with me a pre-print connecting z equals zeros with the interval sieve, which we'll come up later in the talk. So everything written with the word new is something new, which I'm going to talk about in this lecture. So we'll start with what's known about large gaps between consecutive primes. There's an upper bound, which is a little bit worse than square root of x due to Baker-Harman and Pence. You can improve this a little bit, assuming the Riemann hypothesis, this is due to Cromere, but that's the best it's known, even conditionally. I think you can do something with the log if you assume RH plus something about the pair correlation conjectures. So there's work of Goldston, Heath Brown, and others on that. But again, this is about the best we know how to do, even conditionally. Well, the best lower bound is a little bit bigger than log x. In recent work I have with Green Conyagin, Conyagin tau. So this fraction does go to infinity, but very slowly. So what is the truth? Well, the first person to make a guess at the truth was Cromere. And I'll talk about how he arranged this conjecture based on a probabilistic model of prime. So he thought that G of x should grow like log squared x, at least in the Lin soup sense, Granville modified Cromere's conjecture, overcoming some flaws that it has, which I'll talk about later. And same order, but his prediction was a slightly larger constant. So how does that compare with computations? The computations show that at least up to 10 to the 18th, the ratio here is, well, it doesn't quite reach one and it certainly doesn't get above one. And you can see these more clearly on this graph. So the horizontal axis is a log squared scale, which turns these predictions into straight lines. So the black step function is actual data of G of x. The green line is the asymptotic for Cromere and the red line is the asymptotic lower bound that Granville predicted. So you see there's, it's not quite reaching these. It looks like at least it's growing at a rate of log squared x, but the constant is not really evident from the calculations. All right, so the first thing I wanna talk about in detail is this, the connection between gaps and Ziegels zeros, which is something I came up with last year. So the first idea is the Jakob Stahl's function. And this is, this was first done by Vestentius and then improved by Erdesh and Rankin in the, all in the 1930s. So if I let U sub T be the set of integers that have no prime factor less than or equal to T. So this is performing the, say the Sib of Eratosthenes up to the prime T. Then if this set has a large gap, say a gap of size Y, then that implies a gap of size Y in the primes. Well, at least, so if I let Q be the product of the primes up to T, which is about E to the T by the prime number theorem. So between capital Q and two capital Q, there must be a gap of length Y in the set U T because the set U T is periodic mon Q. And since that means that there's a set of Y consecutive integers, all of which have a prime factor less than or equal to T. And since the number itself is much larger than T, the numbers have to be composite. So it's a very simple idea. And in fact, it's the beginning point of every paper that gives a lower bound on prime gaps. Now the second idea I wanna talk about, which the relation is not completely clear yet, has to do with the effect of Ziegels zeros on primes and progressions. So a Ziegels zero is basically a zero of a Dirichlet L function with a real character, which happens to be extremely close to one. So I'm not gonna say exactly how close, but the gist of it is if we know that this zero exists then primes in progressions to where the residue class is a quadratic residue or where the character is one. So in particular, the number one will satisfy that, are going to be very deficient in primes. So depending on the size of Delta as a function of Q, the right hand side here will be much less than what you would expect, which is X over phi of Q log X. And X, so a theorem of Gallagher says that you get something like this as long as X is at least a fixed power of Q. So we know by Ziegels theorem that Delta cannot be extremely small. It must go to zero no faster than any fixed power of Q. Unfortunately, the constant implied here is ineffective. So that's one of the biggest unsolved problems in number theory is to make this effective or improve it or show that Delta has a very specific lower bound. Anyway, so the theorem, if we assume that this zero exists is a lower bound for G of X for a particular X. So you take X to be, so the exponential of a power of Q, and then you'll get something like this. Now, without the one over Delta, this is worse than what we know, but of course now depending on the size of Delta, this could be better than what's known unconditionally. So let me give you some examples. That's the theorem I was talking about that gives a lower bound on G of X infinitely often if there is the Ziegels zero. So for each Delta, for each Ziegels zero applied to the modulus Q, there's a particular X for which then G of X is large. Here are some examples. So if Delta is smaller than a negative power of log Q, we get a lower bound infinitely often of log, double log to the power K minus one. And already if K is two or larger, this is better than what is known unconditionally. So if you write it, the unconditional bound in this way. And if we have a really bad zero, so of this type where epsilon Q goes to zero, let's say extremely slowly, then we'd have a lower bound of log X to the one plus Delta where Delta goes to zero extremely slowly. So again, way better than what we can do unconditionally. All right, and the, so go to page seven. So again, there's the theorem. Not really going to go over the proof in detail, but the idea is that it's a very short proof. Once you have the tool that the primes in the progression one mod Q are very small, then what you're doing is via the Jakobsfall idea, you take a residue classes AP satisfying this congruence QAP plus one congruence to zero mod P. And so what happens is if you have a number between zero and Y, which is not covered by these residue classes, then QN plus one has to be prime. So, and then how many primes are there up to X? Well, if the zero is quite strong, this is small. And then there's a very trivial argument for covering the rest of these numbers by the primes in here. So one prime for each of the remaining uncovered. So then what we get is that we have a set of residue classes for the primes up to T that cover the interval zero Y. And for an appropriate T and Y, this will give a lower bound on the prime gap function. Okay, so I don't want to, if you don't understand the whole slide, that's fine. I will be posting these together with the talk later. The idea is it's very short, it's very short proof. Okay, so now I want to talk about random models, Kramer's model in particular. So let's go to page eight. So Kramer's random model was very simple. So knowing that the prime number theorem says that the density of primes near the number N is like one over log N. So we choose a random set by picking N to be in the set with probability one over log N. And that way our set at least globally will match the known distribution of primes. So what Kramer then proved was that the limb soup of the gaps in this random set have this property that they are of log squared size. And that's the biggest that they get. And he wrote that for the ordinary sequence of prime members PN, some similar relation may hold. Okay, so we interpret that as a conjecture. The proof of his relation here is very simple because these numbers are chosen independently. Then the probability that you have K consecutive numbers not in the set, in other words, they are random composites. So this is about one minus one over log N to the K which is approximately E to the minus K over log N. Now you plug in K being a constant times log squared N, this will be N to the minus that constant. So if the constant is one plus epsilon, this is at most N to the minus one minus epsilon. And summing over N you get a convergent sum. If on the other hand you have one minus epsilon log squared N, this is at most, this is greater than greater than N to the minus one plus epsilon, the sum of these diverges. And now you finish up with the Borel-Kenteli lemma which is, let's say the big machine that makes this work. So that's essentially how the Cremère model works. Okay, let's go to page nine. So the Cremère model has good points, bad points and ugly points. The good point, as I mentioned before, it was designed to match the global distribution primes. And in fact, it's a very easy first and second moment argument to show that the number of elements of the random set up to X satisfies the analog of the Riemann hypothesis. Okay, so that's something we expect to be true for primes. And that's good. However, if you look at some local statistics, say the count of twins that differ by two in our random set, it gives you an asymptotic of the shape. However, for primes, Hardy and Littlewood predicted similar asymptotic, but with a different constant. Which is a bit larger. Okay, so Cremère is giving the right order, but the wrong constant. On the other hand, if you look at gaps of size one, well, the Cremère model doesn't really care what the gap is. It's gonna give you the same asymptotic. So this is quite different than the behavior for primes. And so I'll call that an ugly side of Cremère's model. Right, page 10. So in fact, Cremère's model gives that for any finite set of numbers, the number of tuples that appear in the Cremère random set is asymptotic to X divided by log X to the size of the set H, okay? It doesn't see how to say the arithmetic information that is present in the Hardy-Littlewood conjectures. So by contrast, Hardy-Littlewood conjectured the following formula for tuples of primes, where this German S H is the so-called singular series, which is usually nowadays written as a product, but originally it was written as an inference series. So one thing you can notice is that there is some arithmetic factor. So the size of H mod P, that is number of residue classes occupied by H mod P plays a role in this formula. So let's look at some examples. So if H is a singleton zero, then H mod P is one for every prime and these two factors cancel and you just have identically one for the singular series. And then Hardy-Littlewood's theorem or conjecture reduces to the prime number theorem, which is about the only case where we actually know that this is a theorem, literally only non-trivial case. Okay, so H is zero comma two, so that would be twin primes. You compute the singular series because this fraction here is one over two when P is two and it's two over P otherwise. When you have zero one, of course the singular series is zero. Why? Because if you look at the prime two, the set occupies both residue classes mod two, so there's a zero factor in the product. Yeah, and of course then the Hardy-Littlewood conjecture is in fact trivial in that case. All right, let's move on to page nine. That was just a review of the Hardy-Littlewood conjectures. Now I wanna talk about Granville's modification of Kramer's model and this was designed to rectify the problems that Kramer's model has locally with tuples, with twins and triples and so on. So his idea was pretty simple. You do a pre-siving on your set. So T is a little bit smaller than log X and then we choose our random, our numbers to be in the random set. Well, if N is not in UT, we don't choose them at all. They're never going to be chosen. And if N is in UT, which means all of its prime factors are greater than or equal to T, then we choose it with a factor divided by log N where the factor is designed exactly to capture the density of real primes, right? So we know that the real primes live in U sub T, most of them with very small exception, right? Or at least the primes in X2X will all live in this set. Okay, so by doing this pre-siving, we're getting a little bit closer to actual primes. So we're putting a little bit of real arithmetic information into the random choice. Okay, and then it's a relatively easy calculation to show that in fact, Granville's set satisfies the analog of the Hardy-Littlewood conjectures for any fixed set H. Okay, so Granville's model then rectifies these problems with Kramer's model, which are local problems. And since gaps between primes is a local phenomenon, we would really want any random model to have local statistics which are what we expect. So he proved in his paper in 1995 that the gaps in this model are infinitely often a little bit larger than what was predicted for the Kramer model. And the idea is pretty simple. So if Y is a constant log squared X, which is about T squared, so remember, T is given here in this above, then if I look at the elements of U sub T that lie in the interval zero Y, well, that's essentially doing the sieve of aritostinies. You're sieving up to about the square root of Y. So what's left is essentially just the primes plus a little bit of junk, which is negligible. On the other hand, if I took a random choice for A and I just said, what's the typical size of U T intersected with an interval of length Y? Well, because the density is theta, that should be about theta Y elements in that set. And that by Mertens is a constant times Y over log Y where the constant is bigger than one. So what this means is that there is a special starting point, there's a special interval of length Y, which is somewhat deficient in elements of U T and that's then used as a starting point for Granville's argument. Let's move down to perform a kind of probabilistic reinterpretation of this model, so of this conjecture. And I do that by looking at the two factors in the singular series separately. And the first thing we do is we truncate the singular series at a prime Z. Okay, since the singular series converges rather rapidly, if Z is large enough, this is a very good approximation. And then we look at these two pieces. Well, the second piece in blue, you just apply Mertens to that. The first piece, I'm going to interpret as a certain probability. So it's the probability that the set H lies in S sub Z where S sub Z is gonna be a random set. And it's actually similar to what was used in Granville's model, except we're doing things randomly now. So we take a random residue class for each prime and then with all of these fixed for various values of Z, we'll let S sub Z be the integers that don't lie in any of these residue classes, at least for the primes up to Z. Okay, so you can think of this as a random set. So we're not sieving out the zero residue class as we would in a normal sieve, but we're sieving out by a random residue class module H prime, okay? And then it's actually very easy to see that this red product here is equal to the probability that H lies in S Z because H mod P is exactly, or H mod P divided by P is exactly the probability that H intersects one of the, that H intersects AP mod P, okay? And since we want the non-intersection, that's the one minus for that probability. And then there's a trick that goes back to a suggestion that Paulia made in the 1950s. And that is to take Z, well, if you're sieving by primes, usually people think of taking Z to be squared X, but that gives a wrong prediction as well-known and people work in sieve theory. But if you take it to be X to the one over E to the gamma, then what happens to this product in blue? The product in blue becomes just log X to the power H. And then notice that then that will cancel the denominator here. And then what we'll get is this interesting probabilistic reinterpretation of the Hardy-Littlewood conjectures themselves. So the left-hand side is like the probability that a random N up to X, that all these numbers are prime. On the right side, we have now this purely probabilistic expression, which, and there's no singular series anymore. The singular series is gone, which is going to be really important for us later. Okay, go on to page 13. So here I'm taking a pause to simply compare the evolution of the attitudes toward probability theory, at least using probability theory in serious mathematics. So this is a quote from Hardy and Littlewood from their actual, the actual paper where they introduced their K-tuples conjecture. So they wrote, probability is not a notion of pure mathematics, but of philosophy or physics. Well, decades later, the theory of probability was put on much firmer ground. And in the paper that Polja introduced this X to the one over E to the gamma, he wrote essentially defending the choice that for the main reason that it works. So it works, therefore we're going to use it, and then we'll try to find an explanation for it later. And then he makes some, how to say, a colorful comparison here with physics and buying sugar or salt in the grocery store. All right, so now let's move to page 14. And this is now the new random model that I introduced together with Bill Banks and Terry Tao. So we'll first define the Z of X a little bit more precisely by just taking the largest Z for which this Merton's product is at least one of the log X. So then you do have the asymptotic that we want here. So our random set has a very simple definition. It's a set of integers such that N is not in the random sifted set, S, Z of N. Okay, so for each number that we're choosing, we're sieving up to essentially N to the power one over E to the gamma. So let's just, as a sanity check, look at the probability that a given number is in our set. Well, it's given by this expression, which is given by this expression, but which is very close to one over log N because that's the way we designed it. Okay, so that's good. It matches the primes. It matches Kramer, matches Granville. The only problem is that if I take two different numbers, N1 and N2, the events of these being in our random set are very much dependent, okay? And it's this dependency that leads to all the difficulties when we're doing calculations. On the other hand, we claim that the primes and this random set share the similar local statistics. Now, by that I mean K correlations, K tuples, gaps, things like that. All right, page 15. So to state some of our theorems, I want to introduce the notion of a uniform Hardy-Littlewood conjecture for an arbitrary set. Doesn't have to be primes, just an arbitrary set. Okay, so the function Y governs the length or the width of the K tuple and capital K is an upper bound on the number of terms and C is governing the error term. Okay, so there are various conjectures out there that the primes satisfy this with, let's say the width up to root X, K up to log X and with essentially square root cancellation, okay? So there are various conjectures that appeared in the literature. The computational evidence is pretty good for these at least in special cases and what I mean is the square root cancellation on the error term. So for our random set R, we proved that with probability one, we have this Hardy-Littlewood condition not only for individual values of the tuple H, but uniformly. So the width could be a power of X to a power of log X. The K can be as much as a power of log X and then we have some exponent here which is always less than one. In other words, we have power savings all the way up to when C is close to one, we're still getting power savings in our Hardy-Littlewood expressions. Okay, so this is very strong and I haven't written it here but Granville's model does satisfy the Hardy-Littlewood conjectures in some uniformity but it's very weak and the error term only has a logarithm savings as opposed to a power savings. And that's essentially because in Granville's model you're not sieving up very far, you're only sieving by primes up to about log X and ours were going up to a power of X itself. Okay, next slide page 16. So I want to now bring in a connection that Gallagher made between the Hardy-Littlewood statistics and gaps between primes. So he showed that if you assume the Hardy-Littlewood conjecture where the size of H is fixed and in some interval of say zero to log squared X, then he showed that the gaps between consecutive primes essentially have this exponential distribution. So the primes themselves suitably scale behave somewhat like a Poisson process. The main tool in Gallagher's theorem is an average estimate for the singular series over all tuples of a given size in a given interval. This was later improved by Montgomery and Sound who gave some further terms in the asymptotic here. However, the formulas are very poor, they have poor uniformity in K, which is actually really important for our work. And I should mention that Gallagher's argument works for any sequences. In other words, so if you make the same assumption for any sequence, you'll get a similar result for the gaps in that sequence. Now, page 17, we proved a very general result that says if we have an arbitrary set that satisfies the Hardy-Littlewood conjecture in a certain range of uniformity, so we want a width of log squared X. The K is bounded by, okay, kappa log over double log where kappa is some, it could be either fixed or going to zero, so here's the range, there's a big range of kappa that's available. So kappa is determining how good our error term is. Then the large gaps in the sequence are infinitely often at least kappa times log squared divided by log log, okay? So that's for any sequence, it applies to the primes, it applies to randomly chosen sequences, any sequence you want. There's nothing random or probabilistic here. So in particular if K is fixed, then the right side here is almost the size of Kramer's conjecture for primes. So here in this corollary, we suppose that for primes, we have a uniform Hardy-Littlewood conjecture where the K in the K tuple is as large as log over double log, and with a mild power savings, then we'd have this bound here, okay? Again, there's nothing, there's no model here, this is just assuming the Hardy-Littlewood conjecture in a certain range of uniformity, then we get lower bound for gaps between primes, which is almost the size of Kramer's. So we're pretty happy with this because this goes way beyond what you could do with Gallagher's techniques, which I'll talk about on the next slide. Another corollary, because we know from the previous slide's theorem that our random set satisfies a uniform Hardy-Littlewood conjecture, that implies that the gaps in our random set are at least a size log X to the two minus little o one. Okay, unfortunately, this roundabout way of obtaining the gaps in our sequence through the Hardy-Littlewood conjectures is not the most efficient way. So we can actually say more about the gaps in our random set, which I will talk about very shortly. Okay, page 18. So this is briefly a rough sketch of how you obtain gaps in an arbitrary set from the Hardy-Littlewood conjecture. And what's interesting about it is even though it's about an arbitrary set, even a deterministic one, the probability, probability of our random set makes a role, plays a role. And the reason is because it's, well, it's very easy to first reduce by inclusion-exclusion to a sum, a multiple sum over sort of Hardy-Littlewood K-tuples sum here. But now we're going to reinterpret that as a probability instead of writing down the asymptotic in terms of the singular series. And that then greatly simplifies life because we can then reinterpret, again, throwing away error terms, reinterpret this as a certain expectation, which by the binomial theorem is just the probability that our random set has no elements in the interval zero Y. Okay, which is something that we can work on computing. All right, so let's now go to page 19. And this brings up a question which was asked in previous versions of this talk. We know that our random model, or sorry, we know that any set that satisfies sufficiently uniform Hardy-Littlewood conjectures has large gaps of size, almost log squared X. Can we obtain any upper bound on large gaps? I mean something better than, say, square root of X. And the answer is no. And the reason is actually very simple. So if you have any set A that satisfies the Hardy-Littlewood statistics and you simply remove the inner, so the integers from a square root Y length interval for, let's say, a very sparse infinite sequence of Ys, that's not going to affect the Hardy-Littlewood statistics at all. It'll just be absorbed into the error term, but we've created then a very large gap. So we can't do anything about upper bounds on large gaps starting just with Hardy-Littlewood. Okay, page 20. So the last thing I want to talk about is what kind of gaps our new random model produces. Now it's not quite as simple as with Kramer's model because we have this dependence. So I want to connect it with something called the interval stif. So with Z being just a little bit less than square root Y, and that little bit less is important, I'll define WY to be the minimum over all choices of residue classes AP of zero Y intersect SZ. Okay, by the Chinese remainder theorem, if that's a little too complicated, you just look at the next line. This is the minimum over all starting points U of the number of integers in an interval of length Y that have no prime factor at most Z. Okay, so that's an interval sieve. Now, because Z is a little bit less than square root Y, we can use the linear sieve bounds of Yvagnets to obtain a lower bound for WY. We also have a trivial upper bound which comes from taking U equals zero or to take all the APs equals zero. That's the same thing. And that's just, we're just counting primes essentially. Now notice there is a gap between the upper and lower bounds. So we have Y over log Y is an upper bound, essentially Y over log squared Y is lower bound. What's the truth? Well, there's a folklore conjecture that Y over log Y, the upper bound is true. On the other hand, this preprint that Andrew Granville sent me just a few days ago, shows something quite interesting, which is that if Z equals zeros exist, so getting back to the topic I started the talk with, then infinitely often, the WY will be little O of Y over log Y. And in fact, depending on the strength of the Z equals zero, you could make this quantitative. But I don't think that it's possible to reach the lower bound just because we're right at the cusp where the linear sieve bounds just start to become positive. But yeah, so that illustrates the difficulty of pinning down what WY is. I mean, at the very least, we have to solve the Z equals zero problem, which has been around for eight or nine decades now. Okay, next slide. Now we get to the punch line. So I'm gonna let G sub R of X be the largest gap between elements in our random set, which are less than or equal to X. Okay, WY I defined before. G of U is now a new function, which, well, it has a funny definition, but since this WY is going to infinity, you can see at least as it exists, the existing bounds on WY show that G of U grows at least linearly and at most a bit faster than linear. U log U over double log. Okay, so with these definitions, we prove that our random set has maximum gaps that are essentially little g of two e to the minus gamma log squared X, okay, where this epsilon can ignore it if you want. So we use a lot of tools from number theory and large deviation in qualities from probability. And incidentally, as a result of our analysis, the same bounds hold for Granville's random model, okay? So his random model also gave bounds of this type. So now go to the last slide, okay? So here's the theorem and plugging in the bounds that I showed you for the little g function. So we would actually conjecture that we can remove the epsilon here, but we don't know that the little g function behaves nicely locally, so we have to leave it like that. But conjecturally, it would behave like this and the little g function is at least U and possibly slightly larger in magnitude. And that gives us predictions for the largest gap between primes, which are at least of size two e to the minus gamma log squared X, that's the lower bound that Granville gave. And an upper bound, which is a factor double log over two triple log larger. And our, okay, so there's no uncertainty here except the uncertainty is in the answer to the interval sieve problem. So once we know the answer to the interval sieve problem, then our prediction is something specific about G of X. So this opens the possibility that the gaps between primes could be a little bit larger than what, even a larger order than Kramer predicted. And in fact, if you assume Ziegels zeros exist, then our prediction, well, our model then has gaps of larger than log squared X in order, and we predict the same would be true for the primes. So somehow this now, comes back full circle to the topic I started at the beginning of the lecture, that if we know Ziegels zeros exist, you can somehow do better with large gaps, okay? So that concludes my talk and I guess we'll open it up for questions.