Booker’s Extension of the Selberg Class

Briefly, the Selberg class is a set of functions $F:\C\to\C$ such that $f(s)$ can be written as a Dirichlet series for $\Re(s) > 1$ and that satisfies a form of analytic continuation, a functional equation, a Ramanujan hypothesis bound on coefficients of the Dirichlet series, and an Euler product formula.

Andrew Booker in [1] has extended the Selberg class in a different way, in the notion of an $L$-datum. In this post, we’ll state Booker’s definition of an $L$-datum, state his converse theorem, and explain his corollary that the completed $L$-function of a unitary cuspidal automorphic representation of $\GL_3(\A_\Q)$ has infinitely many zeroes of odd order.

Definition. An $L$-datum is a triple $(f,K,m)$ where $f:\Z_{>0}\to\C$, $K:\R_{>0}\to \C$ and $m:\C\to\R$ are functions satisfying the axioms:

  1. $f(1)\in\R, f(n)\log^k n\ll_k 1$ for all $k > 0$ and $\sum_{n\leq x} |f(n)|^2\ll_\varepsilon x^\varepsilon$ for all $\varepsilon > 0$
  2. $xK(x)$ extends to a Schwartz function on $\R$ and $\lim_{x\to 0^+} xK(x)\in \R$
  3. $\mathrm{supp}(m) = \{ z\in \C : m(z)\not= 0\}$ is discrete and contained in a horizontal strip $\{ z\in \C : |\mathrm{Im}(z)| \leq y\}$ for some $y\geq 0$, $\sum_{z\in\mathrm{supp}(m)} |m(z)| \ll 1 + T^A$ for some $A\geq 0$ and $\{z\in\C : m(z)\not\in\Z\}$ is a finite set.
  4. For every smooth function $g:\R\to\C$ of compact support and Fourier transform $h(z) = \int_\R g(x)e^{ixz}\mathrm{d}x$ satisfying $h(\R)\subseteq\R$ there is an equaltiy:
    \sum_{z\in\mathrm{supp}(m)} m(z)h(z) = 2\mathrm{Re}\left[ \int_0^\infty K(x)(g(0) – g(x))\mathrm{d}x – \sum_{n=1}^\infty f(n)g(\log(n)) \right].

For each $L$-datum $F = (f,K,m)$ we define an $L$-function
L_F(s) = \mathrm{exp}\left( \sum_{n=2}^\infty \frac{f(n)}{\log n}n^{\tfrac{1}{2} – s} \right)
for $\mathrm{Re}(s) > 1$. Denote by $\mathcal{L}$ the set of all $L$-data. Then $\mathcal{L}$ under pointwise addition of functions is an abelian group, as can be seen by looking directly at the axioms. If $F$ is an $L$-datum then $F$ is called positive if there are at most finitely many $z\in\C$ such that $m(z) < 0$. If $\mathcal{L}^+$ denotes the set of positive $L$-data then $\mathcal{L}^+$ is clearly a submonoid of $\mathcal{L}$.

Let’s consider this definition. As it’s stated, it’s much different than the definition of the Selberg class. Selberg defines a class of $L$-functions based on properties that $L$-functions should have (or at least, some people think they should have), whereas Booker’s definition gives a set of data that gives $L$-functions, and the data itself is in terms of functions with relatively simple ‘analytic’ properties. In practice, the advantage of Booker’s definition is that it’s much easier to work with. On the other hand, it is often more difficult to see that an $L$-function you’re interested in actually comes from an $L$-function from Booker’s definition. Nonetheless, and thankfully so, Booker’s class contains Selberg’s class, which is definitely not obvious from looking at the definitions.

In his paper, Booker gives an fun application of his definition. One such example of an $L$-datum $F = (f,K,m)$ is one associated to a unitary cuspidal automoprhic representation of $\GL_d(\A_\Q)$. Let $L(s,\pi)$ denote the corresponding $L$-function. In this example,
m(z) = \mathrm{ord}_{s= \tfrac{1}{2} + iz}\Lambda(s,\pi)
where $\Lambda(s,\pi) = L(s,\pi_\infty)L(s,\pi)$ is the complete $L$-function of $\pi$. By studying the properties of $L$-data, one might be able to say something about the order of the zeroes of $\Lambda(\pi,s)$. To do this, let $d_F = 2\lim_{x\to 0^+} xK(x)$ and call $d_F$ the degree of $F$. Let $\mathcal{L}_d$ denote the $L$-data of degree $d$ and write $\mathcal{L}^+_d = \mathcal{L}_d\cap\mathcal{L}^+$.

Following the ideas of Conrey-Ghosh, Kaczorowski-Perelli and Soundararajan, Booker proved the following converse theorem:

Theorem [Theorem 1.7, 1]. Let $F\in\mathcal{L}_d^+$ for some $d < \tfrac{5}{3}$. Then either $d = 0$ and $L_F(s) = 1$ or $d = 1$ and there exists a primitive Dirichlet character $\chi$ and $t\in\R$ such that $L_F(s) = L(s + it,\chi)$.

The interesting thing about this theorem is that although $d$ a priori can be a real number, by this converse theorem we know that it can only be $0$ or $1$ if $d < \tfrac{5}{3}$. Booker gives a corollary for $\Lambda(s,\pi)$ for $\GL_3(\A_\Q)$ as follows. If $F = (f,K,m)$ is the corresponding $L$-data for $L(\pi,s)$, then $F\in \mathcal{L}_3^+$. However, we know by the converse theorem that $\mathcal{L}^+_{3/2}$ is empty, and so $F = (f/2,K/2,m/2)$ is not an $L$-datum. By looking at the axioms for an $L$-datum, we see the only possible obstruction to this is that $m$ satisfies $\{ z\in\C : m(z)\not\in \Z\}$ is finite but $m/2$ does not satisfy this. Hence $m$ is odd infinitely many times, or in other words, the completed $L$-function $\Lambda(s,\pi)$ has infinitely many zeroes of odd order!

[1] Booker, Andrew R. $L$-functions as distributions. Math. Ann. 363 (2015), no. 1-2, 423-454 (arXiv/1308.3067).

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