 Thank you for the invitation to the organizing committee. So it's a pleasure to be here. So I'm going to talk about some recent work on Legendre paths, which we will see will be certain path that will be related to the Legendre symbol. And it's another way of viewing character sounds for that character sounds. Okay, so throughout the talk P will be a prime number. And this here is the usual notation, standard notation for the Legendre path module P, the Legendre symbol module P. It is known that the sequence of values of the Legendre symbol behave randomly in short intervals. And this is, for example, illustrated in the following theorem of Davenport from the 30s, which says the following. So take your favorite sequence, epsilon n of plus or minus one. Let P be a large prime and take K. So think of P as having like a hundred digits, for example, and let K be something up to log P. Then if you let P goes to infinity and you look at the probability that the consecutive values of the Legendre symbol. So A mod P up to A plus K minus one mod P coincide with your sequence. The probability is exactly what you expected. Everything was randomly dependent, which is one over two to the K here. So P over two to the K is the number. And this, in fact, or this fact, this randomness was used by a cryptographer in 1988 who proposed to use the sequence of these consecutive Legendre symbols. So once you shift them, let's say by a secret key and the prime P is public to construct a pseudo random number generator, which is cryptographically secure. And I mean that it's difficult to recover the secret key or to predict what's the next values of this pseudo random number generator if you know just the prime P. And in fact, the best recovery attacks algorithms have complexity, which is exponential, something like square P. So it has applications also to cryptography. So here is another result also which illustrates this randomness. This is a result of that important Erdisch who proved the following. So they show that as P goes to infinity, if you take H to be a parameter such that log H is little or log P and H goes to infinity as P goes to infinity. So think of H as being like less than P to the little or one. And you look at this sum, this short sum of the Legendre symbol. So the sum from N to N plus H, you take a N to be uniformly randomly among the integers zero to P minus one with the uniform distribution. Then the distribution of this short character sum here converges in law to a Gaussian of mean zero and variance H as P goes to infinity. What this means exactly is that if you take this sum here and you normalize by the square root of the variance, so square root of H, and you want it to be in an interval alpha beta. So you look at the number of occurrences of this event. So the number of N in zero P minus one of this event and you divide by the uniform probability, then this converges to the Gaussian. And what's interesting here is that if P is, if H is super large, then this converges to the Gaussian is no longer valid and this is a beautiful recent result of Harper. So in the range H bigger than P over log P to the A, this is no longer valid. So here there's a, and we don't know what happens in between. So here is H less than P to the epsilon in here, H is greater than let's say P to the one minus epsilon in between we don't know what happens. So now I will give you a different perspective of looking at these questions and we're gonna concentrate on longer character sums where much less is known. So here for T positive number less as P of T and this is a notation that we're gonna take throughout the talk. This will be the sum up to T of the Legendre symbol module of P. Okay, so a central problem in analytic number theory is to understand the size and the distribution of this character sum either as P varies or as T varies or as both vary. And in particular, for example, if you, we can have strong bounds for this one, T is very small, then we have information about the least quadratic non-residue module of P. So here are simple facts about SP of T. SP of T is a periodic function of T of period P. This is trivial. On the other hand, it's not continuous. So it has jump discontinuities at the positive integers within a period at the integers one to P minus one and it flicks weights a lot. So we will see later that the maximum of SP of T is always bigger than a constant squared P and it's also less than a constant squared P for almost all primes P. So here I will make this what I mean precise because what I mean is that it's less than a constant that depends on epsilon for all primes except a set of density epsilon within the primes. So it flicks weights a lot and it's not continuous. So what we're gonna do is that we're gonna replace the study of the character sum SP of T, the object in which we are interested by what we call Legendre paths which are better analytically. So we're gonna smooth or we're gonna normalize the character sum by squared P and we're gonna make it into a continuous function. That's what we're gonna do and the object we will end up with we will call it Legendre path. So what is the definition? Here's the precise definition. So the Legendre path attached to the point P is a polygonal path whose vertices are just the values of this character sum. So SP at G or G between zero and P minus one we normalize by square root of P to have something roughly bounded and we glue these points together to have something continuous, glue them linearly. So that's why we end up with a polygonal path. I will show you some pictures soon. So that's the Legendre path. Okay, so here are some simple properties of these paths. First it starts at zero, zero and it ends at P minus one, zero. So it starts at zero, zero, that's trivial. And then at P minus one, zero follows simply from the fact that the total sum the complete sum of characters is zero because we have exactly P minus one quadratic residues and P minus one quadratic non-residues among the classes module P. And so we have cancellation. And we also have a symmetry in the path which comes just from the value of the Legendre symbol at minus one. And so we have a distinction between these two classes. And we will see that there is quite a distinction between these two classes when we look at character sum of the Legendre symbol for other facts too. But more precisely in this case if you take the image at P minus one minus J and you compare it to the image of J we have a complete symmetry in the first case and minus it P J in the second case. So in the first case if you extended with periodicity you end up with an even function. So we will say that the Legendre path attached to a prime P is even if the prime is congruent to three module four. And in the second case if you extend it you get an odd function. And so we will see that the Legendre path with the odd is congruent to one module four. So here are some nice pictures of Legendre path. So we start with the first one which is the Legendre path attached to the prime 191 which is a prime congruent to three module four. And here you see that if you extend this with periodicity then you get an even function. So here the path is symmetric with respect to the line X equals P minus one over two. And here is another Legendre path this time an odd Legendre path but with a prime of similar size. So a prime 997 which is congruent to one module four. And again you see here a symmetry with respect to the point P minus one over two zero. If you extend it then you get an odd function. So we'll ask interesting questions about these paths just like you have pictures and you wanna understand them. And it turns out that some of these questions are correspond to longstanding problems about Legendre symbol and character something in general. So the first question is well you have a path that starts from zero and ends at zero and you wanna understand when does it decrease for the first time. And so the path is formed with sums of the Legendre symbol. So the first time it has the value zero then you add the Legendre symbol of one. So you add one. And so it decreases precisely when the Legendre symbol equals minus one. And so this corresponds to understanding the size of the least quadratic non-residue module of P. So here is the second question. How large is the peaks of the path the maximal distance from point on the path to the X axis. So we think of the path as being sort of a mountain ups and down and you can ask well how large is the distance here, okay. And so this corresponds exactly to understanding the maximum of character sums. And you can ask also how is this quantity distributed as P varies. So this is our second question. And as I said, this is equivalent to studying the maximum of quality character sums and in particular link to the polyabinogradofin equality. So the third question is as follows you have a path which you can think of as sort of an iceberg. Visible portion is above the X axis and a non-visible below. And you can ask what proportion is visible? What proportion of the path lies above the X axis? Because it starts at zero and ends at zero. And in this case, this is equivalent to studying the positivity of these sums, positivity of character sums or in our case, sums of the original symbol. And this is a question that was considered by Montgomery in 1974, okay. So here is a remark here that this question is only non-trivial when the path is even because if the path is odd you get an odd function and so the proportion is just 50% once P gets large. So we're gonna study this question only for even paths. Here is another question. So you have a path and you can ask, well, it starts from zero and ends at zero. It's a graph of a certain function. And you can say what is the number of X intersets for the general path or what is the number of zeros? And this is more or less equivalent to something interesting which is the number of sign changes of quality character sum. So more or less equivalent because you can get the rare event that you know you touch the X axis without crossing it but we will be interested about the number of crossings or the number of sign changes of these character sums. And finally, the last interesting question is as follows. So you have these collections of paths. So these nice pictures for many primes and you can say, well, is there like a law? Is there something that is behind all these distributions? So if you take a very large family of primes, for example, primes up to Q or in the adic interval, as we will see and let the parameter go to infinity, do you get a nice limiting distribution? And for example, if this is the case, then you can ask, well, how random is this distribution? So perhaps it's a Brownian motion. She's like a basic example of random walks. But in this case, we will see that we do indeed have a nice limiting distribution but it's not Brownian motion. It's something different. Okay, so let's start with, so we have these five questions. Let's start with them. And the first question is, when does the Legendre path decrease first sign? What is the size of the least quadratic non-residue? So let me tell you a little bit about this question. What's known in the literature? This is one of the most classical problems in this theory. So NP will be the least quadratic non-residue modular prime. So this is smallest n such that the Legendre symbol is minus one. What we know about this, we have, so the first result is due to Vinnogrado from 1918 that you have P to the one over two squared E as a bound for N of P. And the best result that we have is due to Burgess and it's more than six years old and you need to prove the exponent of Vinnogrado to one over four squared E plus epsilon. So there are some results that improve the P to the epsilon but this is the best exponent up to date. And we don't know, we don't have any idea on how to break this exponent but the conjecture is very far, the conjecture due to Vinnogrado is that you should be able to replace this by zero. So we should be able to prove that N of P is less than a constant times P to the epsilon. And it turns out that we can prove this conjecture conditionally. So assuming the drawing stream hypothesis we have even a much stronger bound. So from P to the epsilon we have, we can make it down to log P squared. And this is a result of N kidney from 1951. So in particular GRH implies Vinnogrado's conjecture. And this result is interesting because it has another application to a primality test which proved that primes need P much before the AKS primality testing but it was conditional. So that's what's called the Milner-Rabin primality test or the strong pseudo prime test. And assuming the GRH, one of the steps uses Ankeny's bound for N of P and gives that the test is polynomial. So the running time is bigger of log P to the four log P. So in this question, let me mention a modest contribution of mine joined with Shinan Lee and Sander Rajan from 2015. Assuming GRH we made Ankeny's bound explicit. There was back in Sorenson before us who had NP less than two times log P squared and we make it less than log P squared for all primes bigger than five. Okay, so that's the first question. The first time that the Legendre path decreases. Okay, so now let's see what do we know about the second question which is the size of the peaks. So you have a Legendre path like this and then you wanna know what is the size of the peak? How large can the Legendre path go up or down? So you take the absolute value to study the distance and I said the quantity we wanna study this distance is exactly this. So you maximize over all T's of the character sum up to T and then you normalize by squared P because you want to smooth things. You don't want big fluctuations, okay? And because the complete character sum up to P is zero, you can restrict your attention to the integers between zero and P minus one. So what do we know about this quantity? So we have the trivial bound that this guy is bounded by P. So if you divide by squared P, you get squared P but in fact, we can do much better. And this is the classical Polyabino-Gradoff inequality from 1980 that M of P is bounded by log P. So the unnormalized character sum is bounded by squared P log P. On the other hand, we can show it's an easy argument just using Parseval's theorem and combine it with a Polyafory expansion for character sums that I will show you later in the talk allows you to prove the existence of an absolute constant, even an explicit one, I think pi over 12 should be admissible such that for all the primes, we have M of P is bigger than a constant. And so in summary, M of P is always between a constant and log P for all the primes P. And we wanna understand what is, how large can this guy be and what is the distribution of this guy? That's what we wanna understand, okay? But first, we can ask the following question. Can we improve the Polyabino-Gradoff inequality? And it turns out that this is a difficult question. So the result I'm gonna show you which is a recent result due to Manjorel, some version of it was proved by Bober and Goldmacher. But this result, what it tells us is that if you, you have M of P is less than log P. So if you are able to change this to little of log P instead of less than a constant over primes congruent to three, module four, then you can prove Vino-Gradoff's conjecture for the least quadratic non-residue. The conjecture that I showed you before and for which we don't have any clue on just making the exponent of bird is one over four squared E down to zero. So this tells you that improving Polyabino-Gradoff is gonna be super tough. Of course, there are some instances where we can improve Polyabino-Gradoff inequality, for example, for characters with an affixed odd order due to Grand Ville and Saint-Roy-Jean or for characters with a smooth conductor. So, but for the specific case of the Lejeune-de-Simble and module four primes congruent to three, more four, it's extremely difficult. So then we can ask, well, maybe it's optimal, but it turns out that the answer to this question is no. So if you assume the John's Dremel hypothesis, then you can go down from log P, which is the Polyabino-Gradoff to log log P. And this is a result of Montgomery and Bourne from 1977. So assume the GRH, then M of P is big O of log log P. And why also it's interesting? So it's interesting because we improved, they improved the Polyabino-Gradoff inequality, but it's also interesting because this order here is optimal. And this is a much older result. It's due to Palais. So the construction of Palais works for the Chronic-Rissimble, so characters with attached to fundamental and quadratic characters attached to fundamental discriminants. And we need Linux theorem bound in the least prime arithmetic progression for it to be extended to primes or to the Legendre symbols. And in this case, you can show that there exists a constant, positive constant in an infinite sequence of primes, such that M of P is bigger than a constant times log log P. And so assuming GRH, we answer the question that the maximal order of M of P is around log log P. But then we can ask, well, what is this constant here? Now, what's the best constant? So let delta be the limb soup of M of P divided by log log P as P equals infinity. And we can ask, well, what is the value of this limb soup? So Joshi in the 70s extended results of Bateman and Chowlock quadratic characters to the case of prime discriminants to this case and show that delta is bigger than e to the gamma over pi, where gamma is the order matroni constant. And on the other hand, Gramblin-Samburajan in 2007, refined the conditional result of Montgomery and Vaughn and showed that delta is bounded by twice this constant, twice e to the gamma over pi. And here this is another instance where, you know, there is a discrepancy with a factor of two between the omega result in this case and the all results given by the GRH, another related problem is the bounds conditional on GRH for the maximum of L functions on the critical line, where the, and again, there was a discrepancy of two between what we can prove for omega results in this case in the exponent and what the GRH gives you. And in these problems, we believe that it's the omega results which is closer to the truth and it's the same in this case here. So that's a conjecture of Gramblin-Samburajan from 2007 that the Limsub should be e to the gamma over pi. Okay, so conjecturally we should be able to, so conjecturally m of p, the maximum of m of p is e to the gamma over pi plus little or one omega p, but we can prove it. And so in order to understand which of these two bounds is closer to the truth, one way to do this is to study the distribution of m of p and that's what we're gonna do. So let me recall that m of p is between a constant and log p, that's what we have unconditionally for all primes. And Montgomery and Vaughn in 1979 were the first to study the distribution of this quantity m of p and they show that m of p is bounded for 99.99% of the primes, okay? Or if you wanna be more precise, if you put any epsilon, you specify any epsilon to be small, then there is a constant with depends on epsilon such that you have this inequality m of p is bound by this constant for all primes except a set of density epsilon within primes. And so the event that we saw that m of p can grow up to log log p is very rare. It's a very rare event, most often m of p is bounded. But the Granville sound conjecture tells us that m of p should grow as large as e to the gamma or pi log log p and that this is the maximum. And in fact, Granville sound made more precise conjectures depending on the congruence of the prime modulo form. So the conjecture is that you should get larger values if you are three mod four, larger by a factor of square three. So you can get Joshi's constant e to the gamma over pi if you are three mod four, but only e to the gamma over square root three pi if you are congruent to one mod four and these are the maximum constants. And the goal of this study is to understand this conjecture and prove some results that you can get that will actually support it. So we're gonna study the distribution of very large values of m of p, which we're gonna do. So in order to have some nice results and not have this constant in the distribution function, we're gonna just normalize by it. So take m of p and divide by e to the gamma over pi multiplied by pi over e to the gamma and call this quantity little m of p. And we are interested about these two quantities. So we want the proportion of primes up to x, which has congruent to one mod four and such that this quantity is bigger than v. And we also want to understand the proportion of primes up to x, congruent to three mod four and such that little m of p is bigger than v. Okay, so I'm gonna remind you of these definitions. Eunice, there is a question in the chat from Henrik Vanjic. So I could read it. Yeah, yeah, please. Should you have said the value of the L function at one rather than the critical line? Probably refers to the previous slide. No, no, on the critical line too, because, okay, let me go to the previous slide. Okay, all right. Yeah, so on the critical line on GRH for the Riemann zeta function, we know that on GRH, a log of zeta half plus IT is bounded by log T over log log T. While the best omega result we have is that the maximum of zeta half plus IT in the dyadic interval, let's say, is bigger than exponential of square root log T. So there is a discrepancy of two. So square root log T in the bottom in the omega result and log T in the top. So that's why I mean the two here. I don't know if that... Yeah, okay, can I say something? Yes, yes, of course. Is there any relation? If I remember correctly, Montgomery and Born had this constant for E to gamma over pi. Yeah. I from estimations for the L function at 0.1. Yes, yes, you're right. So, but how... It is connected, yes, you're right. So what happened is that... Okay, sorry, need to go. I mean, that's how you cannot also improve easily for your Vino Gradov because it... Yeah, indeed. So everything is related. So M of P is... So this guy is, in fact, bigger than L1 kind. Okay, thank you. Yeah, so whenever you have an omega result, then you have an omega result for M of P. That's, yeah. So for example, this, to my, yeah. So this result of Josh, you can recover it if you can show that there are infinitely many legendar symbols such that L1 at the legendar symbol is bigger than E to the gamma over pi. So he proved it differently, but you can do that. Yes, it's related. Okay, so let me go back here. So we are interested about these quantities. So the proportion of prime such that little M of P, which is just bigger M of P normalized by this constant so that you have clean result, let's say bigger than V. And we have a discrepancy between these two cases, one mode four and three mode four. So the result I showed you before of Montgomery and Vaughn that M of P is bounded for almost all primes, you can, you can extract the following bounds from their work. So again, these are, so I remind you of these, these are the proportion of primes you are studying and what they showed is that these proportions decrease faster than any negative powers of P. Okay, so that's the result, but only for fixed V and this is important. So the motivation for us is that we want to understand super large values of M of P. And so we want to find, that's the goal, we want to find an estimate for these two distribution functions uniformly for V up to what we expect the maximum is, which is the Granville sound conjecture. So for P up to X log log P is roughly like log log X. So the maximal region here is V up to one minus epsilon. Let's say log log X and similarly in the other case of one more four, where you have the same, but you divide by square root of three. Okay, so that was the motivation of our work and we could in fact provide such estimates. So, so this is a recent result, which was on archive last fall up to the optimal range or almost up to epsilon. Okay, so this is optimal up to epsilon. We have a nice estimate, precise estimate for these distribution functions. So for primes congruent to three mod four, it decreases double exponentially, exponential minus exponential of V plus some error term. And for primes that are congruent to one mod four, you can see this square root three factor conjectured by Granville and sound origin in this estimate. So, so here are two conclusions. So first of all, because this distribution here, this tail decreases much faster than this one, we can draw the conclusion that almost all primes that have super large redundant symbol sums must be congruent to three mod four. So the three mod four, let's say wins in this case. And also because we get the exact constant, so one here and square root three here, this gives a strong support to the Granville sound origin conjecture that the maximum should be exactly this. So M of P, the maximum should be, so capital M of P should be E to the gamma over pi, log log P if you are congruent to three mod four and you need to divide by square root three if you are congruent to one mod four. And in fact, if our results were to hold in a slightly bigger range, change this minus epsilon to plus epsilon, then the conjecture of Granville sound origin were to follow. So this is the second question. Okay, so maybe let me tell you some remarks here on this result. Bob or Goldmacher, Granville and Cuckolopoulos proved a similar result, an earlier one, for another family, which is a family of non-principled directly characters modulo life prime Q. But here, let me stress out that the proof is very different from ours because their proof relies heavily on the orthogonality relations of characters. So we have a certain sum of characters for which you need to bound super large moments and I mean moments up to log Q or Q is the prime here. And the orthogonality relations of characters tells you that only the diagonal terms contribute. But if you have other families of characters, for example, ours, then this proof does not generalize. So we discovered a different approach which is more flexible and allow us to handle other families, in particular family of characters associated to fundamental discrepancy or the family with prime discrepancy, so family of Lejeune de Symbols, the result that they showed you. Now, in this case, there is a slight difficulty coming from possible single zeros that we need to address and that's why our result is not as precise. So for example, here we have this error term here. And the method relies principally on the quadratic large step. So we combine several inequalities in particular Heath Brown's Seminole paper from 1995 on the quadratic large step and an inequality of Montgomery and Vaughn just specific to prime discriminants from their page on a 79. And you are actually working on generalizing the method for this distribution of cubic character as well. And so this will be treated in a future project. So the sieve also is working nicely in this case. Okay, so let's talk about the last three questions. A bit faster than before. So the first one is, or the third one, remember you have Lejeune de Path. Think of it as being like an iceberg with a visible part above the x-axis and a non-visible part below the x-axis. And you wanna know what is the proportion of the visible part, okay? So we're gonna focus only on even Lejeune de Path which corresponds to the case three, four, we call SP of t, that's the character sum up to t. We can ask a nice question which is how frequently is SP of t positive? This is a question that Montgomery asked in 74. And so we're gonna measure the points t for which this is true, okay? So let's normalize by p here. So instead, we're gonna look at t's in zero one and we would like to understand this quantity lambda p which is the Lebesgue measure of points in zero ones that's that once you normalize by p, the parameter t here, you have positive, okay? So SP of tp is positive. What did Montgomery, so that's the proportion of the path above, strictly above the x-axis, okay? So Montgomery showed that no matter what prime you take, you always have 2% visible. So lambda p is at least one over 50 for all primes, uniform, which is a very nice result. And then he showed that you can construct an infinite sequence of primes such that the proportion of visibility is less than a third plus epsilon. So two thirds below the x-axis and one third above the x-axis. And he also noted, he didn't prove this, but he noted that one can use similar ideas to show that there are infinitely many primes such that almost everything is above the x-axis. So there is nothing below or maybe epsilon below. So lambda p is bigger than one minus epsilon. So in a master thesis in 2005, when he studied these questions, she slightly improved the constant one over 50 by, you know, an epsilon. Most importantly, she run extensive numerical computations which suggests that the proportion one third in Montgomery's second result is best possible, that you cannot get less than, then you cannot get more than two thirds negative. And okay, so the extreme case is one third, positive and two thirds negative. In the other extreme cases, one is everything positive. That's what Montgomery did. And she was also interested into changing infinitely many here to a positive proportion of primes. So she proved this result, assuming the dry stream on hypothesis, for a positive proportion of primes, the proportion of positivity is less than 0.746. Montgomery has less than the third plus epsilon by just with an infinite number of primes. And she showed that for a positive proportion of primes, congruent to three month four lambda p is bigger than 0.285. Okay, Montgomery had infinitely many, but one minus epsilon. So using my work on distribution of large character sums that I showed you before. So some ingredients in this work, we were able to improve both results to in fact, change the words infinitely many into positive proportions in Montgomery's result. So for any epsilon, there is a positive proportion of primes for which you have everything, lambda p is bigger than one minus epsilon, almost everything above the x-axis. Another positive proportion of the primes for which you have the either extreme case or conjectured extreme case, two thirds negative and one third positive. And we can also quantify these proportions in terms of this guy, epsilon. And you can also allow epsilon to go to zero as big as infinity. For example, if you're interested into a hundred percent, a real hundred percent of positivity, for example, you can take epsilon to be a small negative power of log p and it's true in this result. So let me very quickly in one minute give you the statement of the result. So change epsilon to one over t. So for t up to arbitrary large power of log x, the proportion of primes such that lambda p is bigger than one minus one over t is bigger than this quantity. And the other extreme case, you have log x to the one minus something small, delta is small, and you consider lambda p less than a third plus one over t. And you can get that this is bigger than exponential minus t to the two plus delta. Okay, so in the last few minutes, I would like to talk quickly about the remaining questions. So the first one is, remember how many times the character path crosses the x-axis, which is equivalent to sign changes of character sums. And here I can't escape, I need to use fourth iterates of the logs, which proves to you that this is a true analytic number theory talk. So we're gonna use log k for the k-th iteration of the logarithm. This is again the character sum we saw before. p of x is a set of primes up to x. And l plus will be the set of primes such that sp of t is always positive. So we can view this as a walk, a random walk, and you want it to be always above the x-axis, always no crossing. So I think there are questions in the chat. There are none, it's fine. So there's a question by David Meyer. The truth that there are infinitely prime, number of primes such that lambda of p equals one. Exactly. Equals exactly one. Yeah, that's, I don't know. That's a very good question. Yeah, which is related to this question here because here it's like equals, but like in a stronger sense that never happens. Yeah, the construction, yeah. Yeah, the construction doesn't give you one. There is always one minus something that depends on p. Yeah, but that's an excellent question that I would like to look at in the future. Yeah, thank you. Okay, so here is another related question. So lambda p is one, but in a very strong sense that no crossing, nothing crosses or touches the, or nothing crosses the x-axis. And let L plus be the set of these primes. What we can do is bound them, bound the number of these primes. So Bacon and Montgomery show that it happens with 0% of the time. So the number of primes is close to zero, our proportional primes goes to zero as x goes to infinity. And this was improved recently by Kalminin who were the saving of log log x. So log log x to the power alpha for some small alpha. And as a consequence of some of my work with Olyxic Lurman and Mark Munch on Fickety polynomials, we can prove the following results. So, but the drawback here is that we have to assume that there are no c goes zero. So we have a similar result for the chronicle symbol, which is unconditional, but for Legendre symbols of prime discriminates much more difficult. And in this case, if you assume there are no c goes zeros, then you can improve this proportion to exponential of minus a constant log log x over triple log x. And we, in fact, can say a lot more, not just that the sets where there are no crossing is bounded by this quantity, but we can say that for almost all primes, we have a lot of sign change. And that's the next result. So assume that there are no c goes zeros, then there is an exceptional set of primes, very small. So pi of x with some saving, so exponential of minus triple log to some power a, a is a constant, such that if you're outside this set, so for almost all primes, then the character sum SP of t has many sign changes. So has at least double log, log log divided by quadruple log. So think of it as just double log. If you don't like, if you're not an analytic number theorist, just think of this as log log x sign changes and not only that, but also very early on. So in like less than p to the epsilon, you have many, many crossings of the x axis for a typical prime. But the drawback is that you have to assume no c goes zeros. Again, if you change this to fundamental discriminants, then the result is unconditioned or writing. So we have five minutes enough for the last question, which is the distribution of these paths. So we have a collection of Legendre paths, nice pictures, once p goes infinity, do we have a nice limiting distribution? So we call that the path, this Legendre path is a polygonal path with vertices j, zpj, where zpj is just a normalized character sum or sum of the Legendre symbol up to j. You can normalize by square root p. And you have a path. So let's look at it from a different perspective. Let's look at it as a function. So let fp be a continuous parameterization of this path. And because it's a continuous function, which is periodic of period one, we have a Fourier series expansion for it. And this Fourier series expansion is just an easy consequence of the Fourier Fourier expansion for character sum, because there is a very small difference between fp of t and sp of t. The difference is just big old one over square root p. So it's in this error term. But anyway, so you have this Fourier series expansion and let's not mind this factor here because it's gonna be just a constant. So tau of p is the Gauss sum, which is square root p if p is congruent to one more four and I square root p if p congruent to three more four. So again, you see this script in C between the cases of primes congruent to one more three more four. Here is the Legendre symbol and here you have the complex exponential. So exponential minus two pi i and j. And what we're gonna do is that we're gonna construct to run the model for fp of t by modeling the Legendre symbol using a random random multiplicative function, which is a model which has been studied extensively during the last several years, maybe last decade. So we get a model of the Legendre symbol for m in z for negative integers by non-negative integers by random variables xn, where for positive integers, these are exactly random multiplicative functions, which I just find as follows. So xm is a completely multiplicative function in m with x1 equals one. And the xq's for q prime are independent identically distributed random variables taking the values plus or minus one with probability one half. So that's a good model for the Legendre symbol. And you extend it to negative integers by multiplying by random variable x minus one, which just mimics the value of the Legendre symbol at minus one. So it should be also taking the values one minus one with probability one half, because the primes are equidistributed mode four. And xm minus m will be completely multiplicative. So it's x minus one times a random matrix on the multiplicative function. And so this is our parameterization of the Legendre path. Remember, we had this, this is just the Gauss sum divided by square p. So it's one if p is congruent to one more four and I if p congruent to three more four, this is the Legendre symbol. So the random model will be as follows. So change the Legendre symbol to the rather macro random multiplicative function and then change this AP to y, which depends just on x minus one, which this is just modeling the value at minus one, which is the same as modeling whether p is congruent to one more four or three more four. So in the first case, it's one here as in AP and in the second case is I. So this is the good model for the Legendre path. We can show that this is almost surely a continuous function for your series of continuous functions. And so it makes sense to look at this, once you vary T as a stochastic process, a random process on the space of continuous functions. And what we prove here is that we prove convergence in law. So we prove that limiting distribution is exactly this random series. So that's the result joint with Aisha Hussein as p varies among the primes in a dietic interval. So we have to use a dietic interval because there are some technical issues on the size of prime. And the size of the interval goes to infinity. This, the process coming from the Legendre path converges to the process characterized by this random three series in the sense of converges in law in the space of continuous functions on zero one. So what do I mean by this converges? Well, I mean exactly what follows. So that's the precise statement for any bounded continuous function or continuous map from the space of continuous functions to the complex numbers. We have the following. So the sum of this map on this process Fp, so the average between capital Q and two capital Q if you divide by the total number of primes in this interval converge as Q goes to infinity to exactly what you expect which is the expectation of the off the running process. So this result was proved earlier by Hussain in her PhD thesis assuming the joint Riemann hypothesis and then jointly we were able to remove the joint Riemann hypothesis from group though. It was a bit tedious because there was several parts in particular. There was also annoying SQL zeros here which we need to show that they don't contribute to certain moments of character something. So let me end, okay, my time is up. So I will end with nice pictures. Okay, so let me remind you of this Legendre path attached to this prime of size, probably 1,991 which is concrete to three more four. And here is a sample with 10,000 points of the random Fourier series corresponding to this case. So with X minus one equals minus one. So you can see that these two have similar behavior. And again, in the other case, here is the odd Legendre path corresponding to the other prime 997 which is one more four in this case. And here is a sample of the random Fourier series. And this picture I think is closer to that one than in the case of three more four. So here's the Legendre path and again here's the random model. And that's all I wanna talk about. Thank you very much for your attention.