Alexandra Florea: Simultaneous non-vanishing of L-functions at the central point (NTWS 295)

So, today I will talk about non-vanishing of L functions and everything will be joint work with Hung Bui and Micah Milinovich. So, let me start with a classical question. So, the main question that we're interested in is the following. If we have an L function associated to a Dirichlet character, does that function does that L function vanish at the central point 1/2? So, the center of the critical strip. And there is this general philosophy that if we have an L function, it vanishes at a special point. Usually, we look at the center of the critical strip. In this case, it would be 1/2. It vanishes either for very special reasons or for trivial reasons. So, for example, if you have an L function with root number equal to minus one, the L function will trivially vanish at the central point. But apart from these trivial reasons, if we have vanishing, then the rule of thumb is that it it should happen for a very good reason. So, one example would be given by the Birch Swinnerton-Dyer conjecture. That would be like the prototypical arithmetic reason for vanishing at the special point. And this conjecture says that if you have the L function associated to an elliptic curve, then the order of vanishing at the central point of the L function is equal to the rank of the elliptic curve. In this case, the critical strip is zero to so the central point is one. Okay? So, if the L function of the elliptic curve vanishes at the point one, it vanishes exactly when the rank is greater than zero. So, E has infinitely many rational points. So, in this case, the vanishing encodes some deep arithmetic. Okay? If there are no such meaningful reasons to get vanishing, then one should expect that the L function is non-zero at the central point. And indeed, this is the statement of um Chowla's conjecture, uh which says that L of 1/2 chi is never equal to zero for any Dirichlet character. This was originally uh conjectured by Chowla for um quadratic characters, but you can extend it to any um primitive Dirichlet character. And there has been done a lot of work um towards Chowla's conjecture. I'm not going to mention everything that has been written in the literature on it. I'll just highlight a few results. So, if we look at the family of all Dirichlet characters modulo Q, there is work of Iwaniec and Sarnak who got who showed that more than 1/3 of the L functions, so more than 1/3 of L of 1/2 chi as chi varies over characters mod Q and Q goes to infinity, uh more than 1/3 have this non-vanishing property. And this proportion has been improved over the years, uh for example, by Bui who got roughly 34% and uh um the record for the family of uh characters mod Q is due to Khan, Milicevic, and Go, who looked at characters modulo Q where Q is a prime, and they got roughly 38%. So, if you look at these proportions um of non-vanishing for the family of all Dirichlet characters, they are somewhere in the range 30 to 40%. Now, if you specialize to other types of uh uh characters, for example, the most natural next family to look at would be the family of real primitive characters. Uh there are better proportions in this family. So, uh Sound showed that more than 87.5% of these characters uh have uh non-vanishing central L values. And if you assume GRH with Loukanidis and Snyder, uh improved the the percentage to 93.75%. So, the first four results all compute moments of L functions. The last one computes what is called the one level density of zeros. Um and as you can see in this family of uh real primitive characters of quadratic characters, there's the proportions are much better much better than the proportion for the entire family of Dirichlet character characters. And one reason that would explain that is that um this is a family with symplectic symmetry and for that kind of family we expect to have some repulsion of zeros from the central point. So, L functions don't want to vanish at the central point and we get better proportions. Uh the question that I'm interested in in this talk is that of uh simultaneous non-vanishing. So, I want to give a bit of motivation uh for studying this kind of simultaneous non-vanishing results. And this goes back to Iwaniec and Sarnak who proved the following results. So, um if you look at holomorphic cusp forms of weight K, uh fixed K and square-free level N. So, N is the parameter which goes to infinity. And uh you fix two real primitive characters chi one and chi two, then a positive proportion of these holomorphic cusp forms have this double non-vanishing property. So, L of half F twisted by chi one and L of half F twisted by chi two are both non-zero at the same time. And this kind of results um has been improved over the years in various directions. So, another result with a similar flavor is a result of Michel and VanderKam who proved a simultaneous non-vanishing result for three L functions at the same time. So, in their work, they fixed primitive characters chi1, chi2, chi3. They are no longer assumed to be real. And then they looked at holomorphic Hecke cusp forms of weight two and prime level Q, and Q is the parameter that goes to infinity. And then they showed that a positive proportion of these Hecke cusp forms of um weight two and prime level Q have this triple non-vanishing property. So, why would we study uh simultaneous non-vanishing? It turns out that these questions are related to um Landau-Siegel zeros. So, in the paper of Iwaniec and Sarnak uh that's uh I mentioned uh before, they compute this average, this moment. So, they sum uh L of 1/2 F * L of 1/2 F twisted by chiD. ChiD is the quadratic character, and they sum over forms either in the weight or the level aspect, and they get an asymptotic formula. And that asymptotic formula has on the right-hand side an L of 1 chiD. So, if you want to get uh to show that there are no Landau-Siegel zeros, so you want to show to get some kind of uh lower bound on L of 1 chiD, then you should show that the L functions on the left-hand side, so uh these two L functions, you would want to show that they are both non-zero a positive proportion of the time. So, we know that each of these L functions is greater than or equal to zero. So, if you can show that the set where both of them are non-zero um has positive measure, then that would give you a lower bound on L of 1 uh L of 1 chiD, and you would be happy. Unfortunately, this approach doesn't work. So, Iwaniec and Sarnak showed that more than 50% of the forms have L of 1/2 F um non-zero and more than 50% have L of 1/2 F twisted by chi D non-zero. So, you can't force these non-vanishing sets to intersect in a subset of positive measure. So, this barely fails to give you a lower bound on L of 1 chi D. There are also results in the other direction. So, for example, Bui, Pratt, Zaharescu and Chek and Matomäki showed that the existence of Landau-Siegel zeros would imply, for example, 100% non-vanishing for Dirichlet L-functions at the central point. Okay, so the main result that we're trying to improve is the this theorem of Zacharias, which is a triple non-vanishing statement for Dirichlet L-functions. Okay, so he proved that if you take Q to be a prime number and you take chi 1, chi 2, chi 3 to be three Dirichlet characters mod Q, then a positive proportion of characters mod Q have the property that L of 1/2 chi chi 1 and L of 1/2 chi chi 2, L of 1/2 chi chi 3 are non-zero. And in fact, he also gives a lower bound on the value of the central values, namely 1 over log Q. Okay, when Q gets large, this goes to zero. So, it's not a very strong lower bound, but in any case, he gets non-vanishing for a positive proportion of characters. He doesn't actually write down the constant that he gets, but if you go through his proof, it should be possible to do it. And what are the main ideas to prove this kind of result? So, he computes modified moments. So, what is the idea behind this? The idea is that um, you want your mollifier to um, this is a Dirichlet series, a truncated series, and you want it to behave roughly like one over your L function. So, um, he defines the mollifier uh, as this Dirichlet series over say L up to L where capital L is some uh, power of Q, and then you have chi of L and uh, mu of L, the Mobius function, and the Mobius function is uh, the one that ensures that you your mollifier behaves like one over L, and then you have some weight here. And what is the main point for this mollifier? Well, you use Holder's inequality, and that gives you a lower bound on this triple non-vanishing pro uh, non-vanishing proportion. So, you get that um, the proportion of characters chi for which L half chi chi J is non-zero for all the J's, that's bounded below by um, this cubic moment here, and by this fourth moment here. Now, if heuristically your mollifier behaves like one over the L function, then the cubic moment that I have uh, on the left-hand side is roughly like summing one, so you should be able you would ideally want to show that uh, this side here is some constant times Q, and the same thing for the fourth moment. Again, if heuristically your mollifier is one over your L function, then it it's like summing one, and then that would also be of size Q. So, if you can show that uh, both the fourth moment and this cubic moment have size Q, then you get a lower bound for the proportion of uh, triple uh, non-vanishing and it will be a positive proportion. The The idea is that these moments grow at the same rate and if you didn't have these mollifiers, then you would have various powers of log Q appearing and that would destroy your positive proportion um result. Okay, so uh now there are basically two parts two independent parts of the proof. The first one is computing this cubic moment. The second one is computing the fourth moment. Uh you see that the fourth moment has absolute value in it and the cubic moment doesn't. So the techniques will be quite different. So for the fourth moment, Zacharias uh computed um obtained an asymptotic formula. So when uh L, the length of the mollifier, is a small power of Q, he computed this fourth moment and he got um some constant. So um So uh this here is a polynomial in 1 over lambda. So this is just a constant. Um and uh to do this to compute the modified moment, he had to first compute a twisted fourth moment. And this has connections to the breakthrough work of Matt Young who computed the fourth moment of L of 1/2 chi with absolute value without a mollifier. And then there has been a lot of uh work on this problem. For example, Hoff considered uh twisted um moments with uh square free conductors. I will just briefly mention these results. We'll go come back to them. Uh there's work of Blomer, Fourie, Kowalski, Michel, Milicevic, Blomer, Milicevic, uh Kerr, Parlinski, Wu and Xi, Wu and so on. Uh Uh, so I'll come back to this fourth uh moment later. Now I want to focus just a little bit on the cubic moment. So, um, as you can see, he also obtains a formula for the cubic moment. And he gets a main term of size uh well, constant size if you normalize and then some error term. So, you see that both of these formulas, if you normalize, you get something which is a constant. So, if you do the ratio, you will get your uh non- non-vanishing constant. So, how do you compute this cubic moment? Well, you start um in the standard way using the approximate functional equation for these L functions. And after you uh apply the approximate functional equation, you end up with two uh terms, a principal term and a dual term. And the dual term um involves averages of this form. So, you need to average the root numbers. Epsilon chi is the root number of LFS chi. Here I renamed my characters instead of chi1, chi2, chi3. Um I let chi chi1 be um chi and then I get chi chi chi1, chi chi2. So, I have a twisted average of root numbers and it turns out that that is related to a hyper-Kloosterman sum. Okay, so then the idea is that you want to uh bound sums of this form. So, you have three L functions, so you have three parameters here and not N1 and N2 coming from the three L functions. Then you have this weight function here V of N0, N1, and N2 over Q to the three halves. So, this roughly means that N0, N1, and N2 is less than Q to the three halves, let's say, plus epsilon. You have the hyper Kloosterman sum. And then you need to multiply by these characters, chi1 bar of n1, chi2 bar of n2. And it turns out that the critical range where you want to understand and bound these sums is the range where one of the parameters n1 is say square root of Q, and then the product of the other two parameters is roughly Q. So I group the two parameters um let's say n0 and then two, that's of size roughly Q. And then I have these averages of hyper Kloosterman sums, which are twisted by sums of this form. So I'm summing over divisors of n, and then I have chi2 bar of n2. So for example, if chi2 is just the trivial character, that sum is just the divisor function d of n, the usual divisor function, which counts the number of divisors. Uh so and there are results due to Kowalski, Michel, and Sawin, where they bound precisely averages of hyper Kloosterman sums twisted by the divisor function. Uh but this is a generalization, so essentially this is what Zacharias had to do. He had to generalize the work of Kowalski, Michel, and Sawin to allow weights which look like the divisor function, but they are not quite divisor function. And all this is using algebraic geometry techniques. Okay, now the question that we want to ask is can we say something about the simultaneous non-vanishing of more than three L-functions? Okay, so we're trying to improve Zacharias' result. Now, if you compute the one-level density of zeros, which is a way of studying the zeros which are close to the central points in the family, then on GRH, Murty and Carneiro, Chirer, and Milinovich, um, in separate papers, showed that more than 1/2 - epsilon of the L-functions are non-vanishing at the central point 1/2. Okay. Now, if you want to use this approach, uh, it's not enough to provide, uh, conditional simultaneous non-vanishing result for L-functions. And why is that? Well, if you let, uh, P, let's say P is the proportion of non-vanishing for L of 1/2 chi, in this case, under GRH, it's 1/2 - epsilon from the work of Carneiro, Chirer, and Milinovich, then, uh, from the proportion of non-vanishing, you can get a lower bound on the proportion of simultaneous non-vanishing just by doing a simple union bound. So, then you would get that the proportion of, uh, simultaneous non-vanishing is greater than one minus four times, essentially, the proportion of, uh, non-vanishing of L of 1/2 chi chi j, which is the same as the proportion in the family, if you, uh, if you re-index your characters. So, then you get that your proportion of non-vanishing is greater than, uh, 1 minus four times, um, the proportion of non-vanishing in the family, which is 1/2 - epsilon, so that's negative, so it doesn't give you anything. So, in order to get a positive proportion of simultaneous non-vanishing, you would want one uh, minus, um, 4P to be bigger than zero. So, you would need, uh, P to be at least three-quarters. Uh, so, we're very far from proving something of this form. So, the one-level density approach doesn't work, so instead, we go back to computing moments, but in a slightly different way. So, uh, the result that we proved is that if you assume GRH, so all of this is conditional on GRH, and we take Q to be a prime number, then we take four characters chi1, chi2, chi3, chi4. Now, they will be primitive characters modulo DI, and the DIs will be small powers of Q. So, we can't go all the way up to Q. We can't get We can't take characters chi1 through chi4 modulo Q. They have to be modulo small powers of Q. For simplicity, we assume that the DI are square-free and pairwise coprime. Then, for Q sufficiently large, we get that a positive proportion of characters we specialize to even characters, uh but that's just a minor technical thing. So, we show that a positive proportion of these characters have um a quadruple non-vanishing property. So, L of 1/2 chi chi j non-zero uh at the same time for all the four L functions. And we also have a weaker unconditional result. So, we have the same restrictions on the DIs. So, they are small powers of Q, square-free, and pairwise coprime. Then, we get that the proportion of um simultaneous non-vanishing is bigger than Q to the 1/3 minus epsilon times the maximum of the Djs to the minus 2/3. So, the Djs again are small powers of Q. So, this gives you an infinite family of characters, but not a positive proportion, because we have roughly a constant times Q characters, and this is um this is not a constant times Q. So, this um the proportion goes to zero in this case, but it is an infinite family. Um now, another question uh that we want to ask, if you uh recall in the um Zacharias result, he was showing that, uh, there uh, there is a positive proportion of characters mod Q so that L of 1/2 chi chi J are bigger than, uh, 1 over log Q. So, this was from, uh, the Zacharias' paper. Uh, for the three, uh, three L functions result. Uh, we actually improve this for four L functions. This is work in progress with the same co-authors and Bojan Mikanovic. So, again, we assume GRH and then we show that there exists a positive proportion of characters mod Q so that, um, this central L values, L of 1/2 chi chi J is bigger than the exponential of, uh, a constant times square root of log log Q. Okay, so, um, this, uh, bound, this lower bound here is to be compared with 1 over log Q. So, this is, uh, these are quite large values of the L functions. Um, and also, we show that we have a positive proportion of characters so that the L functions are, uh, simultaneously uh, small. So, smaller than exponential of minus a constant times square root of log log Q. Okay, so, um, what is the strategy of proof? Well, we go back to computing moments, modified moments, because the one-level density approach doesn't work in this case. So, let's assume that we have a modifier which will be defined later. Then, if we use Holder's inequality, uh, in this form, we will get a lower bound on the proportion of quadruple non-vanishing. So, um, over here we have, uh, I I [clears throat] a plus on the sum, that means that I sum over even primitive characters. So, what I highlighted here in green is the proportion of quadruple non-vanishing for even primitive characters chi mod q. And we get a lower bound in terms of two quantities. So, we have a ratio of something that looks like a fourth modified moment. So, we have a sum over chi mod q of L M chi chi 1 L M chi chi 2. Then we have an L M chi chi 3 bar and L M chi chi 4 bar. I'll come back to explaining why we put the bars on the L function a bit later on. And it's when we apply Holder, we also get uh in the ratio, so in the denominator, we get this uh sixth moment. Sixth modified moment. So, um if we have in mind that approach of Zacharias, he actually computed uh for him he had a cubic moment and a fourth moment. He actually computed the cubic moment and the fourth moment, and he did the ratio. Now, for us this is not going to work. Why? Because we don't know how to compute high moments of L functions. So, for example, we don't know how to compute the sixth moment of L of 1/2 chi when we do absolute value. So, I mentioned the previous work on the fourth moment, the work of um the work of Young and improvements by many other people. But the sixth moment is completely out of reach, and if you want to introduce a modifier, that's even um more unfeasible. So, you could compute a sixth moment if you introduce an extra average over say little q up to capital Q, but we don't want to do that. We just want to sum over characters mod q. So, uh computing this sixth moment here is out of reach. But one observation is that if you want to get a lower bound on the highlighted green proportion of non-vanishing, then in fact you don't need to compute the sixth moment of the sixth modified moment of these L functions, but it's enough to get an upper bound for it. So this is our strategy. We want to define a modifier so that we can prove an upper bound for the modified sixth moment, which would be sharp. So we want to show that this is bounded by a constant times Q. And then we also want to show that this fourth moment here is bounded below by another constant times Q. So if you have a lower bound on the fourth modified moment, and if you have an upper bound on the sixth modified moment, then when you do the ratio, again you will get a positive proportion of uh simultaneous non-vanishing. So this is what we will do. It turns out that uh the first two parts go hand in hand. So um to we define the modifier and we prove upper bounds for the modified sixth moment, but it also kind of goes the other way. If we want to prove upper bounds for modified moments, then that kind of tells us what the modifier should be. And these ideas go back to work of uh Soundararajan, who proved upper bounds for moments of the Riemann zeta function. That uh those upper bounds were almost sharp. They were in Harper, who got sharp upper bounds for moments of the Riemann zeta function. And then there was work of Lester and Radziwill on uh upper bounds for modified moments in uh various uh certain families of L functions. So with that work in mind, so inspired by these kind of ideas, we define the mollifier through an Euler product. So, what does that mean? We're not going to define our mollifier in the way I showed you before in Zacharias work where he has a sum over Dirichlet series and he has the Mobius function which ensures that you have cancellation with your L function. Instead, we start with work of Chandee who bounds log of the L function, log of absolute value of L by a sum over primes. Well, prime powers. So, he she has sum over n up to x which are prime powers. So, this is the von Mangoldt function here which is just supported on prime powers. And really the most interesting are the primes. And then you also get some log Q over log x. So, we're truncating have flexibility in how we truncate the sum over the primes. So, let's say we truncate up to some point x and that introduces a log Q over log x term. So, if you exponentiate this, you will get an upper bound for your L function. Okay? So, the L the L function in absolute value will be essentially bounded by the exponential of the sum over the primes. So, roughly we have that L of 1/2 chi is bounded by the exponential of the sum over the primes. Roughly chi of p over square root of p. I'll put it in quotes. It's not exactly this, but roughly this. You have some weight for the chi of p and you have this extra factor, but roughly you have this exponential of the sum over the primes. And now the main idea is that you can approximate the exponential by a truncated Taylor series. So, if you want to approximate the exponential of T and T goes up to say H over E squared, then you can truncate the exponential of T by the Taylor series truncated at the point H where H is the upper bound on T. So, if you replace your T by the sum over the primes, then you roughly would get that the L function in absolute value is bounded by sums of these forms. So, when you expand everything, you truncate at some point H, let's say. So, now you're summing over N whose primes are all less than X. Let's think of X as being roughly Q to the epsilon. We truncate at the point H, so we sum over integers N with at most H prime factors. So, this capital omega of N counts the number of primes of N. And then we have chi of N. We have this new function U of N, which is just a multiplicative function which um um which controls the repetitions that we get. So, how many times you get this N when you expand when you expand the Taylor series. Um and this is roughly what sums we would have to consider. So, if we keep in mind this um model of Zacharias where he was defining the modifier um as a Dirichlet series with mu of L chi of L over square root of L, then maybe this would uh lead us to say that, okay, then maybe we can define our modifier in this form. So, it would be exactly the same uh series that I have over here. So, we have the same kind of sum over integers whose primes are all less than say Q to the epsilon with at most H primes and then we just throw in this Mobius function here which takes care of cancellations. But it turns out that if you run this argument with an optimal choice for H, so the point where you truncate, this does not lead to a sharp bound for the modified moments. So we have to refine this argument and then what we do is we apply these ideas behind Harper's work on moments upper bounds for moments of the Riemann zeta function where instead we split our primes into infinitely many intervals and essentially we use the fact that primes small primes behave differently from large primes. So we don't want to deal with the small and the large primes in the same way. So we split the primes into all these intervals. It doesn't really matter at this point what the intervals are. And then we stop when say BK is a small constant, so we have like Q to the epsilon. Okay, so we only consider primes up to Q to the epsilon. And then we define our modifier on each of these sub intervals. Okay, so the J'th piece of the modifier will be a sum over integers n whose um primes are all in the J'th interval and they have at most let's say HJ prime factors. These are parameters that you need to choose carefully. And then the modifier has the same structure from before, so you have chi n mu of n over square root of n. And then you define your modifier as the product of all these pieces. There are of course constraints on the parameters, so you want all of these when you expand them, you want to get certain Dirichlet series so that you can actually compute moments of those polynomials. So, you have certain conditions on the beta betas and the H's, but that's a more technical thing which doesn't matter too much right now. Um so, if we if we adapt the method of uh Soundararajan and Westerlund Radziwill, then we get that uh we have um sharp bounds for uh any uh any mollified moments, not just for the sixth moment. We need it for the sixth, but we um can make this work for any um any moment. Okay, so that finishes the uh the discussion about the upper bounds. Now, we also need lower bounds for the mollified fourth moment. And here the techniques are completely different. So, now our modifier is set. We already chose it so that it allows us to prove these upper bounds for uh high mollified moments. Now, once we have modifier, we want to show that we have a lower bound for the fourth moment here. So, notice that we have the absolute value on the moment outside, so it's not in the L function. But, if you look at this moment, for example, if you specialize to chi1, chi2, chi3, chi4 being uh the trivial character, then you actually recover the fourth mollified moment with absolute value. So, although we don't have absolute value in the L function in this average over characters, it kind of behaves like a moment with absolute value. Okay, and there has been a lot of work over the years. I will not mention everything. Um as I said, this uh the first power saving uh asymptotic formula goes back to work of Young. There were improvements by Blomer, Fourie, Kowalski, uh Michel There has been work on twisted fourth moments, twisted mollified moments, shifted twisted moments, and so on. So, a lot of work. Some of these papers deal with Q prime. There's been recent work where the the assumption of Q prime was removed, and so on. So, for us, we want to generalize all these results. So, now we want to consider a moment where in the character we twist by these characters chi one through chi four. So, we want a twisted fourth moment with character chi one through chi four, and the twists are the ones that come from the Euler product mollifier. Okay, so um uh If we want to do the modified moment, we need to do a twisted moment. So, this is the main object that we're interested in. So, we have the product of L of 1/2 chi chi one, chi chi two, chi chi three bar, chi chi four bar, and then we multiply by two twists chi of L1 and chi bar of L2. We start in the classical way. So, we use the approximate functional equation. Uh we get a principal term here. Um and the coefficients are these convolutions of characters. So, chi one star chi two. This is a generalization of the divisor function. So, it's a sum over, let's say, AB is equal to M of chi one of A, chi two of B. Um so, again, when chi one and chi two are trivial, then this is just the usual divisor function. And then, one um one appealing feature Okay, so I should say we also have this weight function here, V. Um which is rapidly decaying. So, essentially, you need a man to be uh less than Q hat squared. So, Q hat here is roughly, let's say that D is the maximum of the DIs, the moduli of the four characters that we start with. So, this is roughly Q times D. So, we have a sum over MN up to roughly QD squared. Okay, so it's like you're truncating. And then you have a dual term here. And this is very different from Zacharias' work. So, in his work, I showed you that the dual term was involving this twisted average of root numbers. And we have the same feature here, but now the root numbers have the the appealing feature that we have an epsilon of chi chi one, epsilon of chi chi two, epsilon of chi chi three bar, and epsilon of chi chi four bar. And chi and chi j have co-prime moduli because Q is prime. So, actually, they split from the Chinese remainder theorem. So, you can write epsilon of chi chi j in terms of epsilon chi times epsilon of chi j. And because we take the bars on these second L values, the epsilon chi and the epsilon of chi bar will combine, and then you don't need to deal with the oscillations of the root number. Okay, so this is very different from Zacharias', where the bulk of the work was going into controlling the oscillations of the root number. For us, we don't have this feature here because we get rid of the root number. Uh so, here, epsilon is just something which depends on the DIs, and it has absolute value one. So, it's not a root number anymore. So, once we know how to deal with this principal term, Dealing with the dual term will be very similar because the dual term is very very similar to this principal term except that you put the bars on chi1 chi2 and the bars on chi3 bar chi4 bar and you need to multiply by this extra chi factor of d1 d2 d3 bar d4 bar. Okay, so if we use the approximate functional equation, we get um two terms, the principal and dual. And now we introduce the sum over the characters. So we need to use orthogonality for even characters and uh the orthogonality relations um isolate the terms with m congruent to plus or minus n mod q. So we will have a diagonal term which comes from m is equal to n and then we will have off-diagonal terms from m is not equal to n. Um so we compute the diagonal term. This is all quite standard. So if we compute the diagonal term, uh what's interesting is that we get a product of four L functions evaluated at one. So recall that we started with chi1 chi2 chi3 bar chi4 bar. So now we get a product of combinations of products of this. So we have chi1 chi3 bar chi1 chi4 bar chi2 chi3 bar and chi2 chi4 bar. And then we have some factors here which depend on the twist. These don't really matter. These are products over p dividing L1 product over p dividing L2. So just some finite Euler products and then we have some function here which is absolutely convergent, so we don't worry about it too much. Okay, so now we want to introduce the mollifier. So keep in mind our main goal, we want to get a lower bound of a constant size for the modified fourth moment. So, when we introduce the modifier, we have the product of the four modifiers, and each modifier is a product of infinitely many pieces. So, if we expand, it kind of looks bad, but it's not so bad. Um there is a lot of structure in the product of the modifier uh in the product of the modifiers. So, you're summing over integers um m and n, and you write them as products of m naught up to m k and naught up to n n k, so that uh the m j and n j are only divisible by primes in i j. Uh recall that we had the modifier the Mobius function in the modifier, so we get mu m mu n, the characters, and then we have these extra um sums here, which again look like the divisor function. So, if chi 1 chi 2 chi 3 chi 4 are 1, again you just get divisor of uh m j and divisor of n j. Okay, so um then when we introduce the modifier and we have the the product of the four L functions, we essentially need to compute this twisted moment, which we just did. And now we introduce this diagonal term from the twisted moment. And uh if we use our formula for the diagonal term, uh recall that we had a product of four L functions evaluated at 1, and then we have some term which comes from the modifier. Okay, with these coefficients, say a p, which is chi 1 plus chi 2, and b p, which is chi 3 bar plus chi 4 bar. So, if we're careful, we can actually write this as an Euler product over P, let's say P less than Q to the epsilon. And um this will be 1 minus So, I need to multiply AP and BP. So, it's something like chi1 chi3 bar over P and then 1 minus chi1 chi4 bar over P and so on. Uh but the L functions, again, if you think of truncating them, then that would be like a product up to Q to the epsilon of 1 plus chi1 chi3 bar over P and so on. So, if you combine these uh together So, if you combine, say, this term with that term um you multiply them, you get rid of the term which involves 1 over P and you only get things of the form 1 over P squared and those are convergent um Euler products. So, this is how we get our lower bound of constant size. To make it precise, we need to use a result of Granville and Sound on GRH which approximates log of L of 1 chi by a sum over the primes. So, essentially, truncated sum over the primes. So, you approximate the L function by uh a truncated Euler product. Okay. Uh so, this is what happens with the uh diagonal term. So, you have these things which are of the size 1 over P squared. So, that's why you get your um lower bound of constant size. But there is an issue. There are, in fact, six main terms, uh not just this diagonal term. Uh there are five more terms, but these, after you introduce the modifier, are negligible. Okay? So, you're only left with the contribution from the modified diagonal, which is bounded below by 1 by a constant. Uh so, you're good. Um so I won't say too much about how to extract these other terms and how to bound your off-diagonal term because I don't have a lot of time. Um I will just say so right now um we have the sum over M and N and we have these uh convolutions of uh characters. Um now we assume that M here is of size uh let's say roughly M and N is of size roughly N. And we use ideas from the work of Young and also from work of Zacharias. So we have two different types of regime, the balanced case and the unbalanced case. So when the the variables are far apart, when M and N are far apart, we follow Young's proof and um that works even when we have these uh more general coefficients, not just the divisor function. Uh when the variables are close together, we follow Zacharias and that's using spectral methods. So we start with the delta method of Duke, Friedlander, and Iwaniec. And um we end up with sums of this form. So we have a sum over M and N of these generalized coefficient times exponential functions. So then we use Voronoi summation on the sum over M and N. We use it twice, once for the sum over M, once for the sum over N. And in fact, we need to prove a Voronoi summation formula because we uh couldn't find one in the literature. So the point is that when we prove Voronoi summation this on the right-hand side, you will get uh three terms. One term involves L of 1 chi 1 bar chi 2, L of 1 chi 1 chi 2 bar, and then a dual sum. So we need to bound the dual sum. and um we have these main terms here and these will be secondary main terms. So, this be essentially what you see these L functions here. So, this is like swapping your character. So, you get certain one swap terms coming from here. So, you swap um So, we start with chi 1, chi 2, chi 3 bar, chi 4 bar and this is like swapping chi 1 and chi 3. You have another one swap term which swaps chi 1, chi 4, another one which swaps chi 2, chi 3 and chi 2, chi 4. So, in total you have four one swap terms. So, you will get a new term which involves a product of L functions, but now your characters are swapped. So, you get four of these. And you get one more term which is a two swap term. Uh this comes from the dual term in the approximate functional equation. So, this corresponds to swapping chi 1 and chi 3 and chi 2 and chi 4. Okay, so we get uh one diagonal, four one swap terms and one two swap term and then we bound the error terms using spectral methods. Um So, that concludes the discussion about the conditional result on GRH. Now, we also want to prove unconditional non-vanishing. So, how do we do it? We don't do modified moments anymore. So, now we start uh with the fourth moment. The pure fourth moment, not a modified one. And if we use um the upper bound for these L functions due to Petro and Young, so you get a cute D to the 1/6. If you multiply by four, you get QD to the 2/3. So, this fourth moment is bounded by QD to the 2/3 times the number of characters with this quadruple non-vanishing property. So, the proportion of non-vanishing is bounded by QD to the minus 2/3 times the fourth moment. So, if you want to show that you have an infinite family of characters for which you have this quadruple non-vanishing property, then you should show that this fourth moment is non-zero. That's all you need to do. Uh it turns out it's not extremely easy to show it because the fourth moment uh is a sum of six terms. So, you know that each of the terms is non-zero, but you don't know that when you put them together, you also get something non-zero. It could happen that they conspire somehow, and when you add all of these terms together, you get zero. So, we need to deal with that. Um so, we have a sum of six terms. Each of the terms is um of this form. So, this is a product of four L functions evaluated at one, and all the other terms involve products of character swaps of these uh L functions. So, uh you have the one swaps and the two swaps and so on. So, you want to show that when you add all these six products of L functions evaluated at one, you don't get zero. So, what we do that So, if we permute the characters, we can assume that the first product of L functions is the biggest without loss of generality. So, then we get that the main term is bounded below by an expression of this form. Now, for R, we do have a lower bound, which comes from the Siegel bound for the L function. It's ineffective at one. And we have this expression which involves the DIs, and it turns out that this is positive for most DIs. Um but there are a few small cases for the DIs where this is negative. So, in that case, we don't get our result. So, then we are left with a finite number of cases, and we didn't really know how to deal with it. So, this reduces to showing that a certain 6 by 6 determinant is non-zero and then we use Mathematica to do it um because the characters are already built in and we know exactly what the coefficients are equal to. And the very last thing that I will say is that building on this work and using uh work of um Bui Evans, Lester Pratt, and Radziwill, we can now revisit the this question of how big and how small can the values L of 1/2 chi chi j get simultaneously. And we have to prove some kind of version of CLT and that allows us to uh get this uh result which says that not only do you have non-vanishing, but in fact it happens uh very often that uh the L functions get simultaneously quite large. So, I will stop here. Thank you.