--- date: 2023-03-19 18:30 aliases: ["Untitled"] --- References: - # Special Values of $L\dash$functions I think $L\dash$functions are awesome, excellent, great, and cool! Let me tell you about them. # Introduction and Motivation What I'm working toward today is some explanation of what is going on in Chapter 8 of Goss. My impression is that this is really a lot of detailed theory-building without very much context, which can make the motivations and analogies difficult to extract. The thrust of the chapter is that we would like to define a good notion of an $L\dash$function in the function field setting, and in order to do so we almost need to rebuild a version of "complex analysis" that works over $\CC_\infty$ instead of $\CC$. As a technical aside, it will in fact be $S_\infty \da \CC_\infty\units \times \ZZpadic$ which serves as the better analog of $\CC$. We'll need notions of *algebraicty* in order to make sense of transcendence results and of *entire functions* to make sense of analytic and meromorphic continuations. At an even more basic level, to make sense of "analytic" functions of the shape $L(s) = \sum_{a\in A} a^s$, we need good notions of exponentials, exponentiation with "complex" arguments, logarithms, and convergence of power series. So we in fact need a bit more than an analog of real/complex analysis -- we'd like to work toward a functional-analytic theory by putting a Banach space structure on $S_\infty$ (or perhaps even a Hilbert space structure, I can't tell if this has been worked out) along with good theories of integration and measures to parallel the real-analytic theory of $L^p$ spaces of functionals. I'd like to spend some time answering the question, "Why put in all of this work just to define $L\dash$functions? What is the payoff?" The short answer is there are many wide-open conjectures in algebraic and analytic number theory over global fields which are *incredibly* difficult but admit relatively straightforward statements in terms of $L\dash$functions. ## The Langlands Conjectures To begin with, these give perhaps the simplest statements (or really, the simplest *consequences*) of conjectural Langlands correspondences. For example, the work of Wiles-Taylor on the Taniyama-Shimura-Weil conjecture (i.e. the **modularity theorem**) proving modularity of elliptic curves gives an equality of $L\dash$functions $$L(E, s) = L(f, s)$$ where $E/\QQ$ is an elliptic curve and $f$ is a certain modular form attached to $E$. One can view this as a first instance of Langlands duality, where one replaces the modular form $f$ with an automorphic representation and the curve $E$ is replaced with Galois representations that "come from geometry" in some sense. Roughly speaking, we have the following: Conjecture[^1] (Langlands): Let $K/\QQ$ be a number field and $G_K \da \Gal(\bar K/K)$ be its absolute Galois group. For any continuous representation $$\rho: G_K \to \GL_n(\CC),$$ there exists an automorphic representation $\pi$ of $\GL_n$ such that $$L(\rho, s) = L(\pi, s).$$ [^1]: Full disclosure, I've necessarily stated this in a very imprecise way to communicate the spirit of these conjectures. For a much more precise statement, see Arthur and Gelbart page 6. ## Special Values ### Langlands More pertinent to this talk, a general feature of $L\dash$functions is that their special values[^2] carry interesting arithmetic, analytic, and/or algebraic information. The simplest example of an $L\dash$function is the trivial Dirichlet series, $$L(\QQ, s)\da \zeta(s) \da \sum_{n\geq 0} n^{-s},\qquad \Re(s) > 1,$$Riemann's eponymous zeta function. As mentioned in a previous talk, some special values can be identified: $$\zeta(1-2m) = -{B_{2m}\over 2m},\qquad m\in \ZZ_{\geq 1}$$ where the Bernoulli numbers $B_{k}$ were defined as coefficients of the Todd function $$\Todd(z) \da {z\over 1-e^{-z}},$$ regarded as an exponential generating function. An interesting claim I found in the literature is that Euler's original proof of the functional equation was found by first attempting to *compute* special values, and cleverly observing relations to Bernoulli numbers at positive and negative integer values. Although it is a mystery to me how one might compute $\zeta(s)$ at negative integer values *without* the functional equation, this perhaps suggests an approach toward the aforementioned Langlands-type conjectures: although directly proving an equality of $L\dash$functions may be difficult, one could feasibly collect evidence by investigating special values on both sides, establishing (possibly distinct) functional equations, symmetrizing both sides to get a common functional equation that works for both, and trying to compute and directly match up special values on both sides. [^2]: A "special value" for $L(s)$ typically means some dinstinguished subset of an extension of the domain. Clasically, for $L(s)$ defined on $\CC$ or a half-plane of convergence, special values might include integers, even or odd integers, half-integers, and so on. ### Zeta regularization Another interesting use that I've been thinking about recently is the use of $L\dash$functions for **zeta regularization**. I'm not sure if this is terribly common outside of analytic number theory, but it is a well-known technique in physics used to assign reasonable values to divergent sums or to define densities. The most well-known of these is the "equality" $$\sum_{n\geq 1} n \da \zeta(-1) = - {1\over 12},$$ using the special value $s=-1$ to assign some finite rational value to this expression. More generally, for any subset $A \subseteq \ZZ$, one could define a partial zeta function $$\zeta_A(s) \da \sum_{n\in A}n^{-s}$$ and define things like the Dirichlet density of $A$ in terms of $\zeta_A(1)$. One could similarly define $$\sum_{n\in A} n \da \zeta_A(-1)$$ to make sense of divergent sums, provided one has the existence of a meromorphic continuation of $\zeta_A(s)$ to all of $\CC$. There are other fun games one could play here, for example defining enriched notions of "regularized density" as ratios of $L\dash$functions $$d(A, s) \da {\zeta_A(s)\over \zeta(s)} \in \CC(s)$$ and considering special values of the form $d(A, n)$ for $n\in \ZZ$. For example, note that $d(A, 1)$ (or an appropriate limit as $s\to 1^+$) precisely recovers the Dirichlet density of $A$, an analytic alternative to its natural density that is occassionally easier to work with analytically. Although I haven't seen the following used anywhere, one can also use this to make sense of expressions like $${\displaystyle\sum_{n\in A} n \over \displaystyle\sum_{n\geq 1} n} \da d(A, -1),$$ although how exactly one might *interpret* these kinds of numbers is a mystery to me. ### Residues and Orders