Wednesday, April 02, 2008

Transcendental L-functions

If you're a mathematician and follow current developments in mathematics, chances are (i. e., non-trivial probability) that you've seen recent news regarding something called "transcendental L-functions". For instance, this:

Glimpses of a new (mathematical) world
A new mathematical object was revealed today during a lecture at the American Institute of Mathematics (AIM). Two researchers from the University of Bristol exhibited the first example of a third degree transcendental L-function. These L-functions encode deep underlying connections between many different areas of mathematics.

The news caused excitement at the AIM workshop attended by 25 of the world's leading analytic number theorists.

There's also a similar press release from the University of Bristol (UK), which is the home institution of the mathematicians who made the discovery: Million-dollar maths prize comes a step closer

The "million-dollar" reference (guaranteed to get the attention of the great unwashed) concerns the prize offered by the Clay Mathematics Institute for a proof (or disproof) of the Riemann Hypothesis.

There have been a handful of blog posts I've come across about this development, such as these:
Unfortunately, none of the above answer the basic question that must be on the mind of anyone who was the least bit interested in the news item to begin with: What the f∗ck is a third degree transcendental L-function?

So far, I've found very little online that answers the question and is even remotely readable by anyone who's not a certificated maven in the field of modern analytic number theory. Including this discussion at the redoubtable n-Category Café. Sadly, the discussion there (counting the comments) consists mostly of plaintive pleas that echo the question stated in the preceding paragraph, and a couple of responses which are very erudite, but quite opaque to all but (at best) a few hundred humans on this planet. (Not that I'm complaining, mind you.)

Happily, I did find one reference, that explains quite nicely, and in a way that's comprehensible to just about anyone who know a modicum of college math, just what a third degree transcendental L-function is (actually, the information seems to come from this PDF):

Whilst the Riemann zeta-function itself is now reasonably well understood, its L-function relatives are not. L-functions are analytic continuations of the more general Dirichlet series:

where ai are complex numbers. Just like Riemann's function, L-functions can be represented by infinite products involving the primes. They also satisfy particular functional equations. Functional equations shed light on the properties of those functions that satisfy them, and for L-functions F(s) the functional equation is:

where q is an integer called the level, d is the degree, and the numbers ri are Langland's parameters.

In the functional equation, Γ(s) is the standard Gamma function (which is an analytic continuation to complex numbers of the factorial function). The integer d (the number of Γ(s) factors in the functional equation) is the "degree" in the term "third degree transcendental L-function". If all of the Langland's parameters ri are rational or algebraic numbers, then we have an algebraic L-function. Otherwise if any of the parameters are transcendental, we have a transcendental L-function.

Well, honestly, I suppose all that may still leave more than a few people gasping for breath. If it's still as clear as mud, but you really want to get down and dirty, you could check out the Wikipedia links in the above description.

But in particular, perhaps you'd just like to see a simple example of an L-function. The simplest L-function of all is none other than Riemann's zeta function, which is quite nicely described at Foxmaths in one of the links given earlier.

I'd love to go into all of this in a lot more detail, but that's a project for another time. Instead, I'll just show you what is perhaps the next simplest example of an L-function, namely Dirichlet's L-function. Dirichlet was actually one of Riemann's teachers, and he introduced his L-function in 1837 (just a bit more than 10 years after Riemann was born). Dirichlet was giving a proof of the infinitude of primes in arithmetic progressions. The proof consisted of showing that the infinite product expansion of the L-function diverged for certain values of its argument, and that could happen only if there were infinitely many primes in any particular arithmetic progression.

Here is the definition of the Dirichlet L-function as an infinite series:
L(s,χ) = Σ1≤n<∞ χ(n)/ns

(The summation index runs over positive integers, and is written as shown for convenience in HTML.) In that definition, s is the function argument (a complex number, though Dirichlet considered only the real case), and χ(n) is a function called a "character", which is defined on the integers and takes values that are algebraic numbers. It has the property that χ(mn)=χ(m)χ(n), and it depends on the particular arithmetic progression.

Like the Riemann zeta function (and all other self-respecting L-functions), this L(s,χ) has a functional equation, which relates its value at s to its value at 1-s. The equation is a little messy to state, and it helps if you first define
Λ(s,χ) = (π/k)-(s+a)/2Γ((s+a)/2))L(s,χ)

Here k is an integer that depends on the arithmetic progression, and a is (1-χ(-1))/2 (which must be 0 or 1). Given that, the functional equation is
Λ(1-s,χ*) = (iak1/2/τ(χ))Λ(s,χ)

The superscript * denotes complex conjugation, and τ(χ) is a complex number called a Gauss sum, which depends on χ.

What all this shows is that the Dirichlet L-function is a first degree algebraic L-function. So now you know.

If you've found this note informative, and if you want to read a lot more about algebraic number theory, check out my series of tutorials on the subject. (It's a work in progress, not nearly complete, but it will eventually get into much hairier L-functions.)

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