Presentation: Asymptotics of Moments of Dirichlet L-series and Infinite Kac-Moody Lie Algebras

This is a presentation I gave for the Student Number Theory Seminar at the University of Minnesota in the Fall of 2019. In it, I discuss current research I am pursuing with Adrian Diaconu at the University of Minnesota.

Abstract

Understanding the analytic properties of zeta and L-functions is critical to algebraic number theory and in particular the Riemann Hypothesis. In the 1980’s an idea emerged that studying averaging families of Dirichlet L-functions gives analytic information about the original family of L-functions. We consider a well-chosen averaging family and discuss conjectures regarding the asymptotic behavior of the family’s moments, the implications these conjectures have for the Riemann Hypothesis, and the computation of polynomials attached to these conjectures using the theory of infinite dimensional Kac-Moody Lie algebras.

Motivation

In Analytic Number Theory we are interested in understanding the analytic properties of zeta and L-functions, i.e, the Riemann Hypothesis. For example, if

\pi(x) := \displaystyle\sum_{\substack{p \text{ prime} \\ p \le x}}1

then the Riemann hypothesis is equivalent to \pi(x) = \text{Li}(x)+O(x^{\frac{1}{2}}\log{x}).
In general, it suffices to to have strong bounds of the underlying L-functions on the critical line. For example: for small \epsilon > 0 and r \ge 1, and T > 1,

\int_{1}^{T}|\zeta(\frac{1}{2}+it)|^{2r}\,dt = TP_{r}(\log{T})+O(T^{\frac{1}{2}+\epsilon})

where P_{r}(\log{T}) is a polynomial in \log{T} of degree r^{2} implies the Lindelöf Hypothesis, and has large applications for the Riemann Hypothesis.

In the 1980’s an idea emerged where it could be useful to consider an averaging family of L-functions to create a multiple Dirichlet series. L-functions alone behave chaotically, but if a good averaging family is chosen the associated multiple Dirichlet series has nice properties and gives information regarding the analytic properties of the original L-functions in the family. It has yet to be made precise which families are nice, and this is in part why the study of multiple Dirichlet series is hard.

Setup

For d \in \mathbb{Z} non-zero and square-free, let

\chi_{d}(n) := \begin{cases} \left(\dfrac{d}{n}\right) & d \equiv 1\pmod{4} \\ \left(\dfrac{4d}{n}\right) & d \equiv 2,3\pmod{4} \end{cases}

where \left(\dfrac{a}{b}\right) is the Kronecker symbol defined by

\left(\dfrac{a}{b}\right) := \displaystyle\left(\dfrac{a}{u}\right)\prod_{i = 1}^{k}\left(\dfrac{a}{p_{i}}\right)^{e_{i}}

where b = u \cdot p_{1}^{e_{1}} \cdots p_{k}^{e_{k}} is the prime factorization of b, and \left(\dfrac{a}{p_{i}}\right) is the Legendre symbol defined by

\left(\dfrac{a}{p_{i}}\right) := \begin{cases} 1 & a \text{ is a quad. res. modulo } p_{i} \text{ and } a \not\equiv 0 \pmod{p_{i}} \\ -1 & a \text{ is a non quad. res. modulo } p_{i} \\ 0 & a \equiv 0 \pmod{p_{i}} \end{cases}.

\chi_{d} is the quadratic character attached to \mathbb{Q}(\sqrt{d}). Define the L-function associated to \chi_{d} by

L(s,\chi_{d}) := \displaystyle\sum_{n \ge 1}\chi_{d}(n)n^{-s} = \prod_{p \text{ prime}}(1-\chi_{d}(p)p^{-s})^{-1} \qquad (\mathfrak{R}(s) > 1).

These are the L-functions in the number field setting. We have similar L-functions in the function field setting. Indeed, if \mathbb{F}_{q} is a finite field of odd characteristic, then over \mathbb{F}_{q}(x) we have an analogous character \chi_{d}(m) = \left(\dfrac{d}{m}\right) for d,m \in \mathbb{F}_{q}[x] monic. The associated L-function is defined by

L(s,\chi_{d}) := \displaystyle\sum_{m \text{ monic}}\chi_{d}(m)|m|^{-s} = \prod_{\pi \text{ monic irr.}}(1-\chi_{d}(\pi)|\pi|^{-s})^{-1} \qquad (\mathfrak{R}(s) > 1)

where |a| = q^{\deg\,a}.

In the number field setting, for r \in \mathbb{N} and D > 0, define the r-th moment by

M_{r}(D) := \displaystyle\sum_{\substack{d \text{ sq.-free} \\ |d| \le D}}L(\tfrac{1}{2},\chi_{d})^{r}.

We would like to study the asymptotics of this average. Often it’s convenient to work with a smoothed version:

M_{r}(D;F) := \displaystyle\sum_{\substack{d \text{ sq.-free} \\ |d| \le D}}L(\tfrac{1}{2},\chi_{d})^{r}F(\tfrac{d}{D}).

for suitably nice functions F:(0,\infty) \to [0,1]; purely analytic methods produced asymptotics for the original moment from the smoothed version. In the function field setting we define the r-th moment as

M_{r}(D) := \displaystyle\sum_{\substack{d \text{ monic sq.-free} \\ |d| \le D}}L(\tfrac{1}{2},\chi_{d})^{r}.

Conjectures

In 2005, Conrey, Farmer, Keating, Rubinstein, and Snaith put fourth the following conjecture for the asymptotics of moments in the number field setting (see [1]):

Conjecture (CFKRS): For r \ge 1, nice functions F and small \epsilon > 0,

M_{r}(D;F) = D\mathfrak{Q}_{r}^{F}(\log{D})+O(D^{\frac{1}{2}+\epsilon})

where \mathfrak{Q}_{r}^{F}(t) is a polynomial of degree \tfrac{r(r+1)}{2}.

In 2019, it was shown by Diaconu (see [2]) that additional secondary terms emerge in the asymptotic formula for the CFKRS conjecture. So, coming up with the correct conjecture for the asymptotic behavior is difficult. The refined conjecture for the number field case is:

Conjecture: For r \ge 4, D \ge 1, nice functions F, and small \epsilon > 0,

M_{r}(D;F) = \displaystyle\sum_{n = 1}^{\infty}D^{\frac{1}{2}+\frac{1}{2n}}\mathfrak{Q}_{n,r}^{F}(\log{D})+O(D^{\frac{1}{2}+\epsilon})

for explicitly computable polynomials \mathfrak{Q}_{n,r}^{F}(t).

The analogous conjecture for the function field setting is:

Conjecture: For r \ge 4, D \in \mathbb{N}, and small \epsilon > 0,

M_{r}(D) = \frac{q^{D}}{\zeta(2)}Q_{1}(D,q)+\displaystyle\sum_{n = 2}^{\infty}q^{D(\frac{1}{2}+\frac{1}{2n})}Q_{n}(D,q)+O_{\epsilon,q,r}(q^{D(\frac{1}{2}+\epsilon)})

for explicitly computable polynomials Q_{n}(D,q).

The takeaway from these conjectures is that the right-hand sides no longer contain quadratic Dirichlet L-functions at the expense of error terms.

The above conjectures are also important for the following reason: often there is an analogy between arguments in the function field setting and the number field setting, albeit the function field setting is drastically easier. It has been proved that if the refined conjecture in the function field setting is assumed then function field Riemann follows. We hope the proof of the refined conjecture in the function field setting will have an analogous proof for number fields, and this will provide some insight on how to prove number field Riemann. My research concerns computing the \mathfrak{Q}_{n,r}^{F} and Q_{n}. The current goal is to compute the Q_{n} because the calculations in the function field setting are easier to compute from a technical standpoint.

Computing the Q_{n}

So how do we compute the Q_{n}? It suffices to consider a multiple Dirichlet series which is closely connected to the moments for the function field setting (setting all s_{i} = \frac{1}{2} makes this expression look very close to the definition for moments):

Z(s_{1},\ldots,s_{r+1}) = \displaystyle\sum_{d \text{ sq.-free}}\frac{L(s_{1},\chi_{d}) \cdots L(s_{r},\chi_{d})}{|d|^{s_{r+1}}}.

The sum in this multiple Dirichlet series runs over the fundamental discriminants which makes its analytic behavior difficult to analyze. Instead, we work with a modified series Z^{\ast}(s_{1},\ldots,s_{r+1}) whose definition we will avoid due to the construction of this modified series being highly nontrivial. Assuming only the meromorphic continuation of Z^{\ast}(s_{1},\ldots,s_{r+1}), it satisfies a group of functional equations

\begin{aligned} Z^{\ast}(w_{i}(s_{1},\ldots,s_{r+1})) &= Z^{\ast} (s_{1},\ldots,s_{r+1}) \qquad (1 \le i \le r) \\ Z^{\ast}(w_{r+1}(s_{1},\ldots,s_{r+1})) &= Z^{\ast} (s_{1},\ldots,s_{r+1}) \end{aligned}

where

\begin{aligned} w_{i}(s_{1},\ldots,s_{r+1}) &= (\ldots,s_{i-1},1-s_{i},s_{i+1},\ldots,s_{r+1}+s_{i}-\tfrac{1}{2}) \qquad (1 \le i \le r) \\ w_{r+1}(s_{1},\ldots,s_{r+1}) &= (s_{1}+s_{r+1}-\tfrac{1}{2},\ldots,s_{r}+s_{r+1}-\tfrac{1}{2},1-s_{r+1}) \end{aligned}

Z^{\ast}(s_{1},\ldots,s_{r+1}) has a simple pole at (\tfrac{1}{2},\ldots,\tfrac{1}{2},1). In fact, all the poles of Z^{\ast}(s_{1},\ldots,s_{r+1}) are reflections of this pole by products of the w_{i} and w_{r+1}. If we can compute all of the poles, standard analytic methods produce analytic information about Z^{\ast}(s_{1},\ldots,s_{r+1}) and hence Z(s_{1},\ldots,s_{r+1}) as well. We can use this pole data to compute the Q_{n} explicitly.

Kac-Moody Lie Algebras and Root Systems

One major drawback is that the group W_{r} generated by w_{1},\ldots,w_{r+1} is infinite for r \ge 4, so we need some powerful machinery to compute the poles. This machinery is the theory of Kac-Moody Lie algebras. A Kac-Moody Lie algebra can be characterized by the following:

  • A generalized Cartan matrix C = (c_{ij}).
  • A set of linear independent vectors \alpha_{i}, called roots, over a complex vector space.
  • A set of linear independent vectors \alpha_{i}^{\vee} called coroots in the dual vector space such that \alpha_{i}^{\vee}(\alpha_{j}) = c_{ij}.

The roots define a root system: a subset of vectors \Phi contained in the \mathbb{Z}-span of the \alpha_{i} which are invariant under a group W of reflections called the Weyl group.

In our setting W_{r} is isomorphic to the Weyl group of the Kac-Moody Lie algebra with generalized Cartan matrix

\begin{pmatrix} 2& & & -1 \\ & 2& & \vdots \\ & & \ddots& -1 \\ -1& \cdots& -1& 2 \\ \end{pmatrix}

Because this generalized Cartan matrix is nice, computing the action of the reflections on the \alpha_{i} is easy. Computing all reflections of the simple pole is equivalent to finding all real roots of this Kac-Moody Lie algebra; given a real root we can substitute the original pole in to get the reflected pole. We would like a closed formula to compute the real roots. Finding such a closed formula is difficult and this is what I’m currently working to achieve. The difficulty is that the roots associated to this Weyl group are of two types: real and imaginary (imaginary roots do not appear in finite root systems). The real roots \alpha are such that there exists a w \in W_{r} sending \alpha to an \alpha_{i} or \alpha_{r+1}. We are only concerned with the real roots, but distinguishing between a real root and an imaginary root by inspection is not an easy task. Indeed, any root \alpha is of the form

\alpha = \displaystyle\sum_{i = 1}^{r}k_{i}\alpha_{i}+k_{r+1}\alpha_{r+1} \quad (k_{i} \in \mathbb{Z},\,1 \le i \le r),

and there’s not much to directly tell us if \alpha is real. Nevertheless, we have an inductive argument to compute real roots \alpha with fixed k_{r+1}. Computing the real roots for fixed k_{r+1} is sufficient, but we would like a closed formula to compute the real roots.


References

[1] J.B.~Conrey, D.W.~Farmer, J.P.~Keating, M.O.~Rubinstein and N.C.~Snaith: Integral moments of L-functions. Proc. London Math. Soc. (3) 91 (2005), no.~1, 33–104

[2] A.~Diaconu and I.~Whitehead: On the third moment of {L}\big(\tfrac{1}{2}, \chi_{\scriptscriptstyle d}\big) II: the number field case. J. Eur. Math. Soc. (JEMS), to appear. Available at arXiv:1804.00690

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