Functional equation (L-function)

In mathematics, the L-functions of number theory are expected to have several characteristic properties, one of which is that they satisfy certain functional equations. There is an elaborate theory of what these equations should be, much of which is still conjectural.

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A function-- and I'm going to speak about it in very abstract terms right now-- is something that will take an input, and it'll munch on that input, it'll look at that input, it will do something to that input. And based on what that input is, it will produce a given output. What is an example of a function? I could have something like f of x-- and x tends to be the variable most used for an input into the function. And the name of a function, f tends to be the most-used variable. But we'll see that you can use others-- is equal to, let's say, x squared, if x is even. And let's say it is equal to x plus 5, if x is odd. What would happen if we input 2 into this function? The way that we would denote inputting 2 is that we would want to evaluate f of 2. This is saying, let's input 2 into our function f. And everywhere we see this x here, this variable-- you can kind of use as a placeholder-- let's replace it with our input. So let's see. If 2 is even, do 2 squared. If 2 is odd, do 2 plus 5. Well, 2 is even, so we're going to do 2 squared. In this case, f of 2 is going to be 2 squared, or 4. Now what would f of 3 be? Well, once again, everywhere we see this variable, we'll replace it with our input. So f of 3, 3 squared if 3 is even, 3 plus 5 if 3 is odd. Well, 3 is odd, so it's going to be 3 plus 5. It is going to be equal to 8. You might say, OK, that's neat, Sal. This was kind of an interesting way to define a function, a way to kind of munch on these numbers. But I could have done this with traditional equations in some way, especially if you allowed me to use the squirrelly bracket thing. What can a function do that maybe my traditional toolkits might have not been as expressive about? Well, you could even do a function like this. Let me not use f and x anymore, just to show you that the notation is more general than that. I could say h of a is equal to the next largest number that starts with the same letter as variable a. And we're going to assume that we're dealing in English. Given that, what is h of 2 going to be? Well, 2 starts with a T. What's the next largest number that starts with a T? Well, it's going to be equal to 3. Now what would h of-- I don't know, let's think about this, h of 8 be equal to? Well, 8 starts with an E. The next largest number that starts with an E-- it's not 9, 10-- it would be 11. And so now you see it's a very, very, very general tool. This h function that we just defined, we'll look at it. We'll look at the letter that the number starts with in English. So it's doing this really, really, really, really wacky thing. Now not all functions have to be this wacky. In fact, you have already been dealing with functions. You have seen things like y is equal to x plus 1. This can be viewed as a function. We could write this as y is a function of x, which is equal to x plus 1. If you give as an input-- let me write it this way-- for example, when x is 0 we could say f of 0 is equal to, well, you take 0. You add 1. It's equal to 1. f of 2 is equal to 2. You've already done this before. You've done things where you said, look, let me make a table of x and put our y's there. When x is 0, y is 1. I'm sorry. I made a little mistake. Where f of 2 is equal to 3. And you've done this before with tables where you say, look, x and y. When x is 0, y is 1. When x is 2, y is 3. You might say, well, what was the whole point of using the function notation here to say f of x is equal to x plus 1? The whole point is to think in these more general terms. For something like this, you didn't really have to introduce function notations. But it doesn't hurt to introduce function notations because it makes it very clear that the function takes an input, takes my x-- in this definition it munches on it. It says, OK, x plus 1. And then it produces 1 more than it. So here, whatever the input is, the output is 1 more than that original function. Now I know what you're asking. All right. Well, what is not a function then? Well, remember, we said a function is something that takes an input and produces only one possible output for that given input. For example-- and let me look at a visual way of thinking about a function this time, or a relationship, I should say-- let's say that's our y-axis, and this right over here is our x-axis. Let me draw a circle here that has radius 2. So it's a circle of radius 2. This is negative 2. This is positive 2. This is negative 2. So my circle, it's centered at the origin. It has radius 2. That's my best attempt at drawing the circle. Let me fill it in. So this is a circle. The equation of this circle is going to be x squared plus y squared is equal to the radius squared, is equal to 2 squared, or it's equal to 4. The question is, is this relationship between x and y-- here I've expressed it as an equation. Here I've visually drawn all of the x's and y's that satisfy this equation-- is this relationship between x and y a function? And we can see visually that it's not going to be a function. You pick a given x. Let's say x is equal to 1. There's two possible y's that are associated with it, this y up here and this y down here. We could even solve for that by looking at the equation. When x is equal to 1, we get 1 squared plus y squared is equal to 4. 1 plus y squared is equal to 4. Or subtracting 1 from both sides, y squared is equal to 3. Or y is equal to the positive or the negative square root of 3. This right over here is the positive square root of 3, and this right over here is the negative square root of 3. So this situation, this relationship where I inputted a 1 into my little box here, and associated with the 1, I associate both a positive square root of 3 and a negative square root of 3, this is not a function. I cannot associate with my input two different outputs. I can only have one output for a given input.

Introduction

A prototypical example, the Riemann zeta function has a functional equation relating its value at the complex number s with its value at 1 − s. In every case this relates to some value ζ(s) that is only defined by analytic continuation from the infinite series definition. That is, writing – as is conventional – σ for the real part of s, the functional equation relates the cases

σ > 1 and σ < 0,

and also changes a case with

0 < σ < 1

in the critical strip to another such case, reflected in the line σ = ½. Therefore, use of the functional equation is basic, in order to study the zeta-function in the whole complex plane.

The functional equation in question for the Riemann zeta function takes the simple form

Z(s)=Z(1-s)\,

where Z(s) is ζ(s) multiplied by a gamma-factor, involving the gamma function. This is now read as an 'extra' factor in the Euler product for the zeta-function, corresponding to the infinite prime. Just the same shape of functional equation holds for the Dedekind zeta function of a number field K, with an appropriate gamma-factor that depends only on the embeddings of K (in algebraic terms, on the tensor product of K with the real field).

There is a similar equation for the Dirichlet L-functions, but this time relating them in pairs:^[1]

\Lambda (s,\chi )=\varepsilon \Lambda (1-s,\chi ^{*})

with χ a primitive Dirichlet character, χ^* its complex conjugate, Λ the L-function multiplied by a gamma-factor, and ε a complex number of absolute value 1, of shape

G(\chi ) \over {\left|G(\chi )\right\vert }

where G(χ) is a Gauss sum formed from χ. This equation has the same function on both sides if and only if χ is a real character, taking values in {0,1,−1}. Then ε must be 1 or −1, and the case of the value −1 would imply a zero of Λ(s) at s = ½. According to the theory (of Gauss, in effect) of Gauss sums, the value is always 1, so no such simple zero can exist (the function is even about the point).

Theory of functional equations

A unified theory of such functional equations was given by Erich Hecke, and the theory was taken up again in Tate's thesis by John Tate. Hecke found generalised characters of number fields, now called Hecke characters, for which his proof (based on theta functions) also worked. These characters and their associated L-functions are now understood to be strictly related to complex multiplication, as the Dirichlet characters are to cyclotomic fields.

There are also functional equations for the local zeta-functions, arising at a fundamental level for the (analogue of) Poincaré duality in étale cohomology. The Euler products of the Hasse–Weil zeta-function for an algebraic variety V over a number field K, formed by reducing modulo prime ideals to get local zeta-functions, are conjectured to have a global functional equation; but this is currently considered out of reach except in special cases. The definition can be read directly out of étale cohomology theory, again; but in general some assumption coming from automorphic representation theory seems required to get the functional equation. The Taniyama–Shimura conjecture was a particular case of this as general theory. By relating the gamma-factor aspect to Hodge theory, and detailed studies of the expected ε factor, the theory as empirical has been brought to quite a refined state, even if proofs are missing.

References

^ "§25.15 Dirichlet -functions on NIST".

External links

Weisstein, Eric W. "Functional Equation". MathWorld.

This page was last edited on 19 September 2021, at 14:56

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