 Thank you for inviting me, and it's my great pleasure and honor to speak in this international seminar. Now today I want to talk something about Krusman's Psalms, who they can be defined directly as following this complete algebraic exponential song. Now here we have normalization because we expect there should exist a certain square root constellations when you take a sum or a. So they were introduced independently by Pongkaki and Krusman in two different situations, and now the Krusman Psalms have become a very important and fundamental tools in number theory, especially in analytics number theory. So for the square root constellations, there is a very basic estimate due to underway. And so in the case of prime moduli, this is a consequence of his proof on the Riemann hypothesis for curves over finite fields. And shortly later, this was generalized to general moduli by Estimani in 1971, 61. So the general moduli is bounded by the wider function. So Krusman's Psalms are important and interesting at least in maybe for two reasons. And one, they can be very powerful tools in many typical questions in analytics number theory. And also they can be very interesting and mysterious which has independent interests. So you can use Krusman's Psalms as important tools. And also you can regard Krusman's Psalms as interesting objects. So this is a very general framework. And now we are in the position to formulate the questions of cuts as indicated in the title. So in this book, Nicolas Katz proposed three questions. And the first one is on the same change of Krusman's Psalms to prime moduli. And he wants to ask what is the density of this site with positive Krusman's Psalms with prime moduli. And the second one is on the equal distribution on this site when p runs over all prime moduli. And the third one is so for the third one, we need a few words. And firstly, we use the Krusman's Psalms to prime moduli as a local factor. And then you take the product for all p not dividing a. So for this one, you can find it is absolutely convergent when the real part of this is bigger than one. So this follows from the way it's formed. So the question is to ask why there is a certain mass form such that this oil product is given by the air function of this mass form with a certain level. So for instance, I mean, equivalently, if this is the case, you can replace this Krusman's Psalm by the fully coefficient of this mass form. At least when p is good. So this is a point. And this is a book of cuts. And also you can you can also find some recent progressives and interesting applications within this framework from this paper by Philip Mishai. So this is a background. And also regarding the second and first questions, you may imagine if you can prove a certain equity distribution, you can you can conclude the same changes of Krusman's Psalms to prime moduli. So there's another a precise conjecture known as a horizontal substand conjecture. So in this situation, cuts considered when A is a fixed nonzero integer, and he makes p run over all of the good primes. So now you are given for any given intro i between minus two and two. And we consider proportional primes such that this Krusman's Psalms falls into the given intro. So when x tends to infinity, the proportion will be converted into this integral defined with respect to the SATA type measure. So you can imagine this one is more or less motivated by the SATA type conductor for electric curves. But unfortunately, the situation of Krusman's Psalms could be much more difficult because we don't know any, at least right now we don't know how to construct analytical tools. For example, we don't know how to construct i functions. So to motivate, now we want to mention a few things about electric curves. And now you are given this curve, it's given by this equation. And we consider the rational points of this curve over our finite fields in the affine space. So now we define this four billion trees in this way. And there's a classical result by Hussey that the four billion trees can be bounded by two times guard p in absolute values. So if you take a normalization that, I mean, for normalized four billion trees can be bounded by two in absolute values. And this bound is very similar to that of Krusman's Psalms. So for instance, if you replace this quantity by a Krusman's Psalms, you can come back to the original product considered by Necruz-Katz. But for this one, if you put another factor coming from bad primes, you can define a Hussey-Weidt function for this curve. And it was conducted that this function, zeta function, admits a morphic combination to the whole complex plane. And also it should satisfy a function equation. And now this is a, it is not conductor and it's a theorem as opposed to the Tanya-Mashurma-Weidt conjecture. So this is on the modularity of the elliptic curves. And that means you can find a suitable homophic cosmophone, such that for all good primes fee, you can characterize the four billion trees but piece for the coefficient of vise. So this is very essentially in the proof of unwells on the Fermat-Loss theorem. And also it is a starting point for the subtle type conjecture for elliptic curves, at least to non-CM elliptic curves. So in this case, we can we can have a very powerful symmetric, we have a symmetric, any symmetric power functions as analytic tools to study the subtle type conjecture. So in the case of elliptic curves, we have analytic tools to study the distribution of the four billion trees. So if we are back to the situation of a Christmas arms, we want to explore the modular structure of a Christmas arms. And then maybe we can construct a certain functions as analytic tools to study the subtle type distribution of a Christmas arms with prime modular. And then as a consequence, you can conclude some same changes of Christmas arms to a prime modular. So this is a big, this is a general picture. And then we want to subsequently we want to discuss the three questions, the three problems of cost one by one. So basically, we first talk about sign changes and then the critical distribution and then modular structures. So on sign changes, we start from these classical results by Krusnesov. And he was able to he was able to give an upper bound for this moment, the first moment of Christmas arms, when you take the sum over so you take the average of the modular over a consecutive interest. So you can compare this one with the trivial bound given by the, I mean the trivial bound given by the ways bound. So you can replace this one by a divider function and the upper bound could be a x times log x. So now we have a power saving. So you can imagine maybe this consideration comes from come from the sign changes of Christmas arms. But there is another concern that if the consideration come from the sign changes, all maybe there do exist many, many Christmas arms of smaller sizes. So this is a serious concern. And if you want to conclude something about the sign changes, you have to study, you have to give some natural information for average Christmas arms with absolute values. So this was done by Fugia and Michel in 2003. And they were able to give a lower positive lower bounds on this average. So you see the other magnitude is very close to x. And you compare this one with a Krusnesov estimate, you can conclude the sign changes of Christmas arms with the integral modular. But differently, the Krusnesov used the best micro-serial of Mohui forms. But Fugia and Michel used a radical homology to prove certain vertical distribution of Christmas arms. So we will mention what is the vertical subset distribution for Christmas arms in the later slides. So combining the two results, we can conclude the sign changes of Christmas arms with integral modular. And then a natural question is to ask how about the case when you take the modular to run over the sparse integers, for instance, in the site of primes. So then we'll come back to the situation of first question of any class cuts. But unfortunately, we don't know how to say about the situation of primes. But there is another deep result by Fugia and Michel. They were able to capture the sign changes for almost a prime modular. So omega n denotes the number of distinct prime factors of n. And then they can prove you see their sign changes. And here because also mu square of n equals to 1 means n is square free. So if n has a large power of primes as a divider, then its valuation of these Christmas arms will be much easier. So we just consider the square free modular. And also the number of prime factors is at most 23. So it's very natural to expect that if one can reduce this constant further, for example, to the limitation, maybe 1, 2, or 3. So if you can reduce this one to 1, you can capture the prime modular. And there are some subsequent improvements by Sivak, Montmachi, myself, and the current record is 7. So you can replace the constant by 7. And here I want to mention another interesting result by Sakitapu and James Maynard. And if they are allowed to assume the existence of lambda zero zeros, they can reduce this constant further to 2. So this is the limitation at least in two aspects. So for the current approach, I mean the many versions of Sivak's 2 is the limitation. And also it is the limitation of the applications of vertical distribution of Christmas arms. So if they are allowed to assume the lambda zero zeros, they can reach the limitations. So this is a my result, and this is what has been proved by Tafu and Maynard. So you see in the upper bound, you have a quadratic function at 1. And if the value is very small, you can obtain a natural estimate for this average or prime modular. But if you want to capture the sign changes, you have to give a lower bound for the average of Christmas arms with absolute values, so which is very similar to the prior situation with integral modular. So but with absolute values, we don't know how to proceed. But if you replace the prime modular by the product of two primes, this was much earlier done by Philippe Michel. And so if you consider the average of this quantity, you can give a certain positive lower bounds with the correct order of magnitude. And then combining the lower bound with the upper bound by Tafu and Maynard, they can produce the sign changes of Christmas arms with at most two prime factors. So that's the point why they can reach the constant two in their proof. Okay, so this is on the sign changes. And the second one is on the equity distribution. So although we cannot prove the horizontal satellite distribution for Christmas arms, and the work of the analog is much easier. This was proved by Niklas Kassin in 1988. So in an opposite direction, now P is a large prime. And then he makes A to run over the multipractic group. And then when P tends to infinity along with primes, so he can prove the equity distribution of the site of such Christmas arms. And the measure is the same as before. So this is a vertical analog of the horizontal satellite's conjecture. And this is the consequence of the lens work on V2. And on the other side, there are some other related sums. For instance, we can define the Thalia sum with a quadratic twist compared with a classical Christmas arm. And there is a very, very impressive work by Duke Freilander and Ivanis. They can prove that this normalized Thalia sum also satisfies an equity distribution. But the measure is quite different. It is a natural Lebesgue measure instead of the satellite's measure. So if you are given a twist, the situation can be much different. And also, there is a function field analog by Chen Liqai and Wen Nenli. So in the function field sightings, they can prove the horizontal satellite distribution. Yeah, so yeah, this is for the equity distribution. And then we turn to the modular structures. So now you are given a suitable mass castle form with this for your expansion. So if you are able to establish the relationships between the Christmas sums and the Fourier coefficients of mass forms, and maybe you can use the width bound for Christmas sums to bound the Fourier coefficients of mass forms. So this is the aim of the celebrated Rhanodian-Petersen structure to bound the Fourier coefficients individually. And in the opposite direction, because there is a very fruitful theory of spectral theory of mass forms, maybe they can be employed to capture analytic information of the Euler product defined by Christmas sums. And then you can use this Euler product as an analytic tool to indicate the distribution of same changes of Christmas sums to prime order. So this is a general philosophy. But there is another concern that is it too optimistic to be true? So now we want to mention impressive work by Andrew Booker. So if you want to find a suitable high mass castle form to characterize this Christmas sum, so the prime meter should be a very, very large. So this was done based on some numerical computation and also some analysis on Christmas sums. But maybe after this inequality, maybe you want to ask how about the situation when the prime meter is much, much larger than this quantity? And also how about the situation at infinity? So we don't know. But from this theorem, we can imagine it is very likely that the answer to the third question of Neoplasma is negative. But on the other hand, in function field analog, it was proved by Cheyenne and Lee, you can really find a suitable high mass castle form such that the eigenvalue can be given by this Christmas sum, but you have to put a negative sign. So this is for the function field analog. So this one is, I mean, so using this modularity, they can prove the, I mean, the horizontal satellite connection distribution in the function field analog settings. So in the function field analog, in the function field settings, the answer can be positive. But now we want to mention a recent work on the original question of cuts. So now you are given a suitable mass castle form. We can always find infinity manuscript n, which is, which are almost a prime and also the free coefficient and the Christmas sum cannot coincide. So in positive or negative, they cannot coincide. So because we are searching almost a prime, this constant is not quite important. And more generally, we can prove this statement. And now eta is an arbitrary real number. And we are given a mass castle form. And we can always find the infinity to the sign changes. So we don't want to see the sign changes. So there are many, many n such that the Christmas sums are bigger or smaller. And also n is almost a prime and also square free. So in particular, when eta is 41 or 81, the constant will be 100. And also, in fact, we can find a uniform constant r to bond the number of prime factors of n. In fact, we can find a uniform constant. So I think the merit of this theorem is that we can, you can choose a general constant theta. And also, even there's no restriction on the size of omega n. I didn't find any existing literature before like this. So in this way, we can characterize the balance between the Christmas arms and the free coefficients to almost a prime modular. Yeah. So, and also in following the spirit of that one minute, we can give another conditional result to reduce the number of constants, the number of prime factors. So if we are allowed to assume the existence of random zero zero in certain way, we can reduce the constant 100 to 7. Yeah. So now, yeah, we have about one half an hour to explain the idea of the proof. And our starting point is a seal method. So seal methods allow us to transfer from primes or almost prime to integers. And then if you take the sum of Christmas arms, after application of seal methods, you have to study the sum of Christmas arms with integral modular. So that means we can apply the spectral theory of automotive forms. So this is a one ingredient in the proof. And also, we have to study the certain equities revolutions of Christmas arms. And we have to use the erratic cohomology to produce the equities revolutions. And in particular, the key observation in the proof is that we want to transfer from the horizontal distribution to vertical distribution. So this one, this are multi, multi productivity as the consequence of Chinese remainder theorem will play an important role. So you see if you sum over R and S, you have a vertical horizontal distribution for this Christmas arm. But now you have a product or two Christmas arms. So for R, you take the sum over S, and now it is in the vertical direction. And also you for as you sum over R, you have, you also have a vertical direction. So the horizontal distribution of this Christmas arm can be transferred to the joint distribution of the two Christmas arms in the vertical directions. So this is a transference of the difficulty of the problems. And to produce the equities revolutions, we have to study such various arms with symmetric powers. So this is a basic idea. And now we want to, now we want to talk about discuss the theorem as a starting point. So firstly, you are given a sequence of negative numbers. And we want to study this average of prime numbers at prime arguments. So so theorem methods can be a possible option to study the upper bound or lower bound for this average. Yeah, now we want to introduce a very convenient approach invented by Sehberg in his study on twin prime conjecture and ghostbar conjecture. So now we, for instance, we consider this average. And now mu square is a mu is a mobile function. And we consider the square field variables. And a n is non-negative. So a n is the sequence and w n is the weight function, which is also non-negative. And how any divider function. So the point is, we want to find, we want to optimize the parameters and also the weight function to produce a positive lower bounds for this average with x tends to infinity. So if this is a case, you can say there is an n between x and 2x such that a n is positive and also tau n is bounded by rho. So then you solve this inequality and you can produce almost the prime set. So this is the idea of the Sehberg. And it proves to be very important, efficient in the work of a GPY. And now also it can run on bounded gaps between primes. So now we want to present our approach. We have this long average. And here we have this one is the Sehberg series. And now we have a new truncated divider function. So alpha, beta, and delta will be optimized later. And psi characterizes the balance between the cross one sum and three coefficients. So this part gives you a cross one sum is bigger or smaller. So and rho of d is chosen in this way. And we don't want to explore, we don't want to divide the details here, because this is due to the theory of Sehberg sieves. And pi theta is defined as a product of small prime numbers. So the theta is another parameter. So this is our starting point. And if we can prove a positive lower bound for this average for all sufficiently large x, we can say there will be some positive for negative psi such that n might be an almost prime. And then you have to solve an inequality on this truncated divider function. So this is a basic idea. And maybe you want to ask why we introduce this truncated divider function instead of the usual divider function. And also why we have to impose this condition restriction. So for this one, you can imagine because there are two variables and using using the definition of this truncated divider function, we can transfer the horizontal distribution of cross one sums to the vertical distributions. So and then you can choose the alpha beta and the data appropriately. And then you can control the vertical distribution of cross one sums very, very effectively. So this is for the truncated divider function. And but why we introduce this restriction. So you can imagine you can imagine in many applications of sieve masses, you have to study you have to characterize the distribution of the sequence in arithmetic progressions. And then you have to use this distribution to characterize the shifting dimension of the problem. But unfortunately, in this problem on cross one sums, you see in absolute values, there is a cross one sum. And for this sequence, we don't know how to characterize the I mean, we don't know how to produce a asymptotic formula for average of this quantity, even when n runs over consecutive integers without this weight, we don't know how to do. So in in this aspect, this is a very extreme safety problem. So but but unfortunately, but unfortunately, we can we can restrict. I mean, if you see a fortunately, we can restrict n to be a product of, of several prime numbers. So in this way, we can produce them equal distribution of a cross one sums produced a positive lower bound for the average of this quantity. But in this way, because we restrict n to be a product of several prime numbers, we have to give some exact explicit evaluations for the several c weight. So because you see row D is a smooth truncation of the model function. If you want to evaluate the this weight, you have to use the inclusion exclusion principle to give everything explicitly. But that would be a very that would be a disaster for numerical computations. So this is why we introduced this condition, because you can take theta is suitably small and n to be take n to be a product of some prime numbers of suitable sizes, such that n is co prime to this product. So that means you just need to consider this one. So then we for this weight function, we only have row one squared. So that would be a very convenient for the evaluations of the separate c weight. And then we arrive at the this average with a cross one sums. So we need to consider the first moment of this difference when there is one. So this is for the lower bound c. And using holders inequality, we can transfer from we need to study the upper bounds of the fourth moment and the lower bound for the second moment. So I mean, for the fourth moment, that would be much easier. And we just explain the second moment. So you see, for the second moment, we have to discuss, we have to explore the positive lower one for this average. And for the first part with cross one sums, we have to use the the equation distribution across one something in the vertical direction. And for the second one, the argument will be much easier because there is a very, very perfect multipractivity for the free coefficients. So the difficulty falls into the control on this cross term. So we want to explore the non-correlations between the free coefficients and cross one sums. But unfortunately, even your sum over constructive integers over n, we don't know how to capture the constellations or the non-correlations between the two objects. So this is another difficulty. But here, but here is another observation that so for this quantity for the free coefficient and also cross one sums at prime arguments, will be 8 over 3 pi on average if the subtle type conjectures can be proved. So this is a consequence of the subtle type conjecture in the horizontal direction. But unfortunately, they are not proved right now. The fortunate thing is that even we can not prove the subtle type conjecture, we can bond this one and this one by some other constants which are strictly smaller than one on average. So in this way, if k is a suitably large, you can see this quantity because n is a product of many, many prime factors. So this can be very small and this one can be very small. So this motive with us to consider this average with absolute values. But in this way, we cannot, we cannot expect any upper bounds of smaller other magnitude. We cannot explore the non-correlations. But because we can choose the k to be very, very large, we can give an upper bound of this average with correct order of magnitude, but with a very, very small scalar because k is large. So this is a point and this can be explained in this proposition. But we want to say the quasi proposition because here we did not explain which domain effectively. But anyway, you see this one is smaller than one and this one is also smaller than one. So if you take k to be suitably large, for example, k is 7, 8, or 9, this one is very, very small. So this is a point. But to prove this upper bound, we have to use equities distribution of cross one sums and in particular, this one can be a very, very easy starting point. So then we have to study the equities distributions of cross one sums. And firstly, we have to give a geometric interpretation of cross one sums in this way, but for the limitation of time, so we just have to give a very quick overview. So firstly, for the equities distribution of cross one sums, we have to study the wear sums of this shape. But for the cuts version of the vertical satellite distribution, he studied this average over the multiplicative group. And then he was a fit formula, you transfer the problem to the computations of co homology groups. So this is a basic idea. And then cuts was able to prove this upper bounds with a square root constellation. And this is, so this is his main theorem on the proof of the vertical, vertical satellite distribution for cross one sums. And the input constant is also absolute. And this, this was proved in 1988. And in, in 1995, Philip and Michelle was able to generalize the work of E plus costs to incomplete into incomplete sites, incomplete intros. And here I can be any intro of loss smaller than P. And he also has a, he also, he also has an upper bound with a square root P times log P. But you want to, if you want to make this one to be non trivial, you have to assume the loss of intro to be a bit larger than square root of P. So when I is a not very short interval, he can prove some equity distributions of cross one sums when a runs over in this interval, runs over this interval. So yeah, so this is a point of this in, this inequality. And so here we, in fact, in our proof, we don't need this in the colleges worry directly, but we put it puts them here just to explain our basic ideas, how to use the article of Margie to prove the equity distributions of cross one sums in a certain shape. So also we need some binary in your forms. When for a prime model, and also barely near forms, who are almost the prime model, so he curious a product of many, many prime factors. And also we have a linear form with general coefficients, but we have another average over the prime model P. So we use such estimates for averages of where sums to produce the equity distributions of cross one sums. Okay, so these are for the lower bound sieve. And now we want to we want to explain what we have to do for the upper bound sieve. So at least we have to evaluate this average with cross one sums, and also with truncated divider function. So after, so you see, there is a convolution in the truncated divider function. And you use a convolution, you can rewrite the cross one sum, you can rewrite n as a product of another two variables. Then you can transfer from the horizontal distribution to vertical distributions. So after some other transformations, and also we also need to generalize the barb and dump from some serum on primes in our symmetry of progressions, we can transfer the first average to something like this. So now you see the cross one sum disappears, but you have another truncated divider function with different constants of alpha and beta. So now we want to choose alpha and beta. So for instance, we choose alpha and beta in this way. And the truncated divider function can be bounded by k to the omega of n in this way. So this is elementary, and then you can replace this for the upper bounds, you can replace this truncated divider function by this divider function. And then you are facing a very classical problem on the evaluations of averages for several segments. So this is for the upper bounds. But in my proof, I can only treat when k is one or two. So when k is larger, for example, three or four or six or the larger, the argument will be more and more difficult. And we don't know how to proceed very easily. So in the end of this talk, I want to mention some related questions. And the first one is that how to how to characterize this one asymptotically with the general multiplicative function. So this one is to replace this divider function, but on average, on average, h of n should be should be some means for example, in many application of same methods, h of n should be can be some other divider function on average. So for instance, when h is two on average or perhaps, so this one is more or less to the evaluations on the binary divider function against the several weights. So we want to we want to explore some asymptotic evaluations for this average with general multiplicative function. So maybe the saddle point methods will work, but at present, it could be an open question. And also, there is another question that on the correlations between the three coefficients and the cross enzymes. So if whether you can prove this this inequality with a very large saving of with with a large power of log x. So maybe using the general theory of a multiplicative functions, we can transfer this question to the to this average of two products, the products of two cross enzymes. So a bar and b bar is a and b are prime to n. And this is a multipractic universe of a and b. So, so even for this one, we don't know how to how to study. And also, I've seen in calculating the in the first half of this talk, we mentioned that if we are allowed to assume the existence of lambda zero zero, we can reduce the constant, we can reduce the number of factors to seven, such that the cross ones are the three coefficients and cross enzymes cannot coincide. But we want to ask what is the limitation of the method. So maybe using more efficient methods, we can also reduce the constant to two, but but I present I don't know. Okay, so this is these are some related questions. And maybe some of some of you can find and they are a bit interesting. And in the end, I want to mention another another related conjecture also for the modularity of cross one sounds. So here we consider this moments written at the moments of cross one sounds written in this way. So sit up sit up P is a cross one some angle. Because you see the the normalized across some some is between minus two and positive two. So you can define an angle to characterize the cross one some and then you take the symmetry of power. And now we have that this moment. And Evans formulated conjecture on the more modularity of this moment. And when K is a one, two, three or four, we don't need any modularity because the evaluation for this one can be elementary. But when K is five, when K starting from start from five, the citation will be more difficult. So but but now, thanks to many, many scholars, the modularity is known for all K, which is at least a five. And when, for example, when K is five, this is approved by two groups, and also six. And here I mean almost almost means we can associate this one with some some to free coefficient of certain homophobic puzzle forms up to some harmless factors. Yeah, and also for case seven or eight, and we study was able to prove the case for seven. And the jointly is Winston, they can he can prove the case for eight. And more recently, the three scholars was were able to prove all of the situations. So maybe you can find more interesting and impressive works from their talks. And in the end, I just want to mention a student of fashion, the subbar. Oh, sorry. So he can, he can prove, he can prove some, I mean, the modularity of the moment for the hyperchrosome sounds of high rocks. So he is Chinese. So you can find some interesting progress from his PhD thesis. Yeah, so this is on the evidence and the texture for the moment, for the, for the more modularity of the moments of Christmas on. Yeah, so, so now we are, we are coming to the end of the talk. And as concluding remarks, I want to say, you may need counting problems in analytical theory, we require estimates for we usually require estimates for algebraic exponential sounds. And in particular, Christmas sounds can provide very typical and important examples. And then you can use algebraic geometry to study analytical theory. So this is the one direction and in the opposite direction, we can maybe we can use some classical methods in analytical theory to characterize the objects in our symmetry geometry. So this is, this is the path of the stock. And also, yeah, this is also the end of this talk. So thank you very much for your attention.