A Family of Rational Zeta Series

My name is Eric, I'm currently working in Busan National University and I want to talk about a family of rational zeta series. The main point of this talk is that we discover an identity that cannot be evaluated on mathematical software. So I wanted to share this kind of mathematics. So first we want to talk about identity. We are going to tell a little bit about how we found it and we are going to give references. So we consider the Riemanncity function. By the way this is defined. You can see that when you evaluate in an integral, this number is very complicated to work with. We know that when you evaluate it on 3Cube is irrational. We know that at least one of these four numbers is irrational, but we don't know which one. And also we know that at least two of these numbers are irrational, but we don't know which one. Still, mathematicians are able to work with these numbers. For example, Goldbach show the following identity. And in general, when you have a series in which the N coefficient is some rational number multiplied by these terms, these series are called rational theta series. So in this case, when the rational number is one, you just multiply one times each one of these terms. This is a rational theta series. And another example of a rational theta series is the series in which the rational number is given by the binomial terms. You know, you have a cell with endpoints. How many subsets of k points can you construct? Let me introduce this notation for this variation of the zeta series. If you just put a equal to one, then the first term is zero plus one. So you get one to the S and you put a equal to two, then the first term is two. So this is a way to remove the tail of actually the beginning terms in the zeta function. So you think about A as a. Where do you start counting? And the formula that we discover is this formula here in terms of this variation of the zeta function. But when you put a equal to 2, we claim that the following happens and this is a rational theta function. So we have an alternating series. The difference between this and the previous identities is that we have this alternating sign. And you can see that this series involves an infinite number of theta values. And we still don't understand those numbers. So we put this left hand side in different software and they all were unable to find the term on the right. Actually they didn't compute the term on the left. We did this on January of 2013. 2023. Oh, this is a terrible type of 2023. So as for January 2023, the left hand side was not computable. And as soon as, oh, our paper is peer review, I will be in contact with the workflow community so that they can integrate this identity. But you know, we first need peer review. Okay, so how we found something that cannot be evaluated in a computer. So what we do is we study operate theory. We want to study operations. But this sounds a little bit abstract. What does it mean to study operations? So we like to use this kind of diagrams trees. And I want to think of this as the addition. So you have your tree and then you have drops of water. And these drops of water on the leaves, when they are pulled down by gravity, they end up in the root. And here I have the total number of drops of water. So these trees represent in addition. But for this to be to make sense, I need the following condition. If here I have three, four and five drops of water, and here I have three, four and five drops of water, at the end, I should have the same total amount of drops of water. This is a way to think about associativity. So if this really represents addition, I need these two trees and variation of this to give me always the same value. The reason why trees are useful for studying operations is because we have a way to encode composition. I don't know if you ever have a garden, but sometimes you have a branch and you can just graft it into a plant and it just keeps on growing. So we use the grafting operation on trees to compose operations. Here I have two binary operations, and then I use grafting to create a ternary operation. So an opera is a family of sets. And you should think of these sets as your nary operations together with composition rules that are associative. In our example, the opera is the operand of trees. So my nary operations are trees with n leaves. And the composition rule is the grafting of trees. What I work with is a way to think about order in an object. And how does ordering an object induces operations that take care of the order. So a poset is a set with a partial order. And one example that you are all might be familiar is when you have programs. Then we say that A is smaller than B if the output of the program A is needed to run the program B. And so apostate is telling you in which way you should order your programs. So you have a chain which is used N programs where the K program requires the output of K minus 1. Also you have two programs and their outputs are not used to run the other one. Then you have A decision union. So there is just two posets and there is no new order. So two programs can be running parallel if they don't require the output of the other program. So this is the case that you have the decision union and you have the ordinal sum of posets. So you forces the maximum elements of P to be smaller than the minimum elements of Q. In other words, the output of the program P is needed to run the program Q. And so if you take this poset where I have three elements and I'm giving these rules, you can write it as taking the disjoint union of A and C and then ordinal sum with B. So I have three programs where the output of A and C is needed to run B. And what we do is we consider the operand of possess, in which my nary operations are trees that have n leaves but every vertex. If the vertex has k children, then I'm going to label my vertex with a poset with k points and in the composable poset on the lexicographic zone. But let me just say a poset. So in this case I have a ternary operation, and you see, I have these vertices that have two children, and then every vertex is labeled with a poset with two points. Now, once you have your operations, the next thing to worry about is where are my operations defined? So given a set, we always have the operand of endomorphisms whose nary operations are functions from X to the N to X. And then an algebra brain operand is a map of operands. But if you go back over the definitions saying that an nary operation becomes a function from A to the N to A. So I'm taking this after operations and they are realized as functions of name. So how does an algebra with an operand look like? So I'm going to give you a geometric example. Given a poset, we construct the order polytope which is a subset of R or 01 to the N in which the points are forced to satisfy the inequalities of the poset. So for example, I have the poset X smaller than Y, and in R squared I'm going to look for the pairs of coordinates where one is smaller than the other and they are between 0 and 1. So this gives me a triangle. If now I have the disjoint unit of two points, then I have two coordinates and they are not related the unit squared. And now if I take X smaller than Y union Z, so Z is 3 and X is smaller Than Y is a triangle. And then, because there is no relation between them, they are just free. So I obtain this geometric object and given to posit the action of this young union on the order polytopes is going to be the Minkowski sum. You take the sum coordinate wise. So here, for example, the order polytope of a point is a line. The order polytope of the poset with one point is a line. The order polytope of this poset is a triangle, and the Minkowski sum is just one plus the other. And you can see by the previous example that given to posets, I have this formula. And this is where something should tell you that there is algebra, there is some compatibility occurring. I can take my posets, compute the order polytope of each one and apply Minkowski sum. Or I can take my posets, compute the decision union and compute the order polytope. The way we think about this equality is saying, you know, on posets there is an action of the operator poset, because I have this is well defined. The operation of the young genome is well defined on polytopes. They are also an algebra over the operator of posets. Because the action of the junior genome is well defined as the Mykoski sum. And there is a morphism that makes all of this compatibility. I can apply one function in one side or the other. This is an operatic homomorphism from possessed to polytops. And actually our theory says that for any poset I have an nary operation that makes this an identity. So we can study operations that understand the order. And what we did in our research is we took posets and we introduced this positive operations. We took polytopes, ordered polytopes, we introduced positive operations, we introduced some polynomials coming from combinatorics, and we showed they also have positive operations. And then we took the zeta values, a variation of zeta values, and we show how they also inherit these operations. The vertical arrow here from polynomials to theta values. We take this as a polynomial on the numerator, So x times x minus 1 all the way to x minus k minus 1 divided by k factorial. And then we assign the following number, the alternating sum that we have here. And using the numerative combinatories and ideas from Ramanujan, we were able to simplify into the identity that we are describing today. And so if these are our main references here, they have some identities, some examples of rational functions. And this is our pre print. As soon as the community verifies everything, I will be in contact with the Wolfram community. To implement this operation. And this is it. Thank you. Oh, there are some questions. Are you aware of the package Sigma by Carson? I am not aware of this package, and I'm gonna test it. Thank you so much.