mixedmath

I'm happy to announce that Eran Assaf, Chan Ieong Kuan, Alexander Walker, and I have just posted to the arxiv a paper on the Fibonacci zeta function \begin{equation*} Z_{\textup{Fib}}(s) := \sum_{n \geq 1} \frac{1}{F(n)^s} = 1 + 1 + \frac{1}{2^s} + \frac{1}{3^s} + \cdots \renewcommand{\Im}{\operatorname{Im}} \renewcommand{\Re}{\operatorname{Re}} \end{equation*} I briefly touched on this (and looked at some visualizations!) in my previous post. I gave a talk on early versions of these results at Dartmouth in 2020.

We introduce a general family of zeta functions and a general family of Fibonacci-like sequences. For a positive square-free integer $D$, let $\mathcal{O}_D$ denote the ring of integers in $\mathbb{Q}(\sqrt{D}) \subset \mathbb{R}$. Let $\varepsilon > 1$ denote the fundamental unit in $\mathcal{O}_D$. We define the $\mathcal{O}_D$ Lucas and Fibonacci sequences in terms of traces of the fundamental unit, \begin{align*} L_{D}(n) & := \mathrm{Tr}_{\mathcal{O}_D}(\varepsilon^n) % \\ F_{D}(n) & := \mathrm{Tr}_{\mathcal{O}_D}(\varepsilon^n / \sqrt{q}), % \end{align*} where $q = D$ if $D \equiv 1 \bmod 4$ and otherwise $q = 4D$. We focus on fields whose fundamental unit has norm $-1$, largely because these have some nice arithmetic properties that make analysis simpler.

The typical Fibonacci sequence is $F_5(n)$ in this notation, as the fundamental unit in $\mathbb{Q}(\sqrt{5})$ is $\varphi = \frac{1 + \sqrt{5}}{2}$. Stated differently, we've defined new Fibonacci and Lucas sequences by their Binet formulas."Binet's formula" was named in honor of Jacques Philippe Marie Binet, though was known much earlier. Stigler's Law!

The $\mathcal{O}_D$ Fibonacci and Lucas sequences correspond to solutions to the quadratic recurrence \begin{equation*} a(n + 2) = \mathrm{Tr}(\varepsilon) a(n+1) - N(\varepsilon)a(n) \end{equation*} with different initial conditions. This is because $F_D(n)$ and $L_D(n)$ are sums of powers of $\varepsilon$ and $\overline{\varepsilon}$, which are the roots of the characteristic equation $X^2 - \mathrm{Tr}(\varepsilon) X + N(\varepsilon) = 0$.

Our main interest is to look at the odd-indexed $D$-Fibonacci zeta function \begin{equation*} Z^{\mathrm{odd}}_D(s) := \sum_{n \geq 1} \frac{1}{F_D(2n - 1)^s} = \sum_{n \geq 1} \frac{1}{\big(\mathrm{Tr}_{\mathcal{O}_D}(\varepsilon^{2n - 1} / \sqrt{q}) \big)^s}. \end{equation*} We also examine the even-indexed one, though I restrict discussion here to the odd-indexed function.

We prove that $Z^{\mathrm{odd}}_D(s)$ has meromorphic continuation to $\mathbb{C}$ in three ways:

1. Using binomial expansion and "rolling up one's sleeves." This is an extremely straightforward argument.
2. Using Poisson summation. For the odd-indexed $D$-Fibonacci zeta function, this is straightforward. We also do this for the even-indexed $D$-Fibonacci zeta function, but the analysis is much more subtle.
3. By recognizing the zeta function as a shifted convolution function coming from modular forms. Unlike the other methods, this is not at all obvious (to us, anyway) and is surprising and interesting.

In this note, I don't dwell on the proofs. Instead, I describe how this project came about and how it connects to my previous research. Superficially the Fibonacci zeta function feels very far from my work, but it's actually deeply connected to much of my work.

Motivation and Context

The path leading to this project was extremely nonlinear. While working on HKLDW20a and HKLDW20b, we were struggling to understand the spectral analysis. My initial efforts were leading to nonsensical statements — I don't quite remember what they were, but they were similar to suggesting that there are only finitely many primitive 3-term arithmetic progressions of squares.

The idea there is to study the multiple Dirichlet series \begin{equation} \sum_{m \geq 1} \sum_{\substack{h \geq 1 \\ \gcd(m, h) = 1}} \frac{\tau(h) \tau(m) \tau(2m - h)}{m^s h^w}, \end{equation} where $\tau(n) = 1$ if $n$ is a square and otherwise $\tau(n) = 0$. We construct that from the inner single Dirichlet series \begin{equation}\label{eq:dh} D_h(s) = \sum_{m \geq 1} \frac{\tau(m) \tau(2m - h)}{m^s}, \end{equation} which is in turn closely related to HKLDW21.^{1 1}which we finished in 2017.

It turns out that $D_h(s)$ has a spectral expansion consisting of only dihedral Maass forms. We were studying these incorrectly and were trying to construct simpler toy problems to understand where we were going wrong.

Coincidences

Two coincidences helped. First, we'd been thinking about the project that became HKLDW19, where we realized that we could count congruent numbers by studying \begin{equation*} D(s, w) = \sum_{m, n \geq 1} \frac{\tau(m + n) \tau(m - n) \tau(m) \tau(nt)}{m^s n^w}. \end{equation*} Shifted convolutions of $\tau(\cdot)$ were strongly on the mind. Second, Tom Hulse attended a talk where Steven J. Miller mentioned that one could recognize Fibonacci numbers by detecting squares.

Chan Kuan always has several students working on several problems and had made some problems similar to this. But for a group of people looking very strongly at shifted convolutions of $\tau(\cdot)$, we look at this proposition and realize that there are integer solutions if and only if the shifted convolution Dirichlet series \begin{equation*} \sum_{n \geq 1} \frac{\tau(n) \tau(5n \pm 4)}{n^s} \end{equation*} is nonvanishing.

This is extremely similar to $D_h(s)$, that we were struggling to understand above. Once we saw this, we began connecting dots.

Generalized Fibonacci numbers

The Fibonacci proposition generalizes^{2 2}Generalizing this proposition in the nicest possible way led to how we defined $\mathcal{O}_D$ Fibonacci numbers.

Consider $\mathcal{O}(\sqrt{2})$ Fibonacci numbers, where the fundamental unit is $1 + \sqrt{2}$ which has $N(1 + \sqrt{2}) = -1$. Thus determining if $n$ is a $\mathcal{O}_2$ Fibonacci reduces to determining if there are solutions to $X^2 = 8n^2 \pm 4$. Any such solution must be even. Dividing out by the necessary factor of $4$ leads to the reduced equation $X^2 = 2n^2 \pm 1$ and the Dirichlet series \begin{equation} \sum_{n \geq 1} \frac{\tau(n)\tau(2n \pm 1)}{n^s}. \end{equation} This is $D_{\pm 1}(s)$ from \eqref{eq:dh}!

Now we saw the connection. This led us to look at the literature where people have considered Fibonacci zeta functions. Approximately every paper involving Fibonacci zeta functions includes a continuation based on the binomial theorem, because it's so straightforward.

More methods of continuation

Originally we thought we might just give the modular interpretation and be done. (Possibly my collaborators would have liked that — it's was mostly me who held us back here). But I wanted to understand why the modular side has such a simplified spectral expansion consisting only of dihedral forms.

It's extremely special. The contributing spectra are evenly spaced on a one-dimensional lattice. Surely there must be some sort of trace formula or something that explains this?

Well, there is a trace formula that naturally lives on lattices: Poisson summation. And Poisson summation very quickly works for the odd-indexed case. (The even-indexed Poisson work in our paper was almost entirely recognized by Alex Walker, and is both very natural in that hand-wavy way and very subtle). But I couldn't make it work out in a way that actually explained why we see what we see on the modular side.

I have no explanation of that even now! But at some point, you must draw a line and call something done.

Different possible directions

In our paper, we craft our story around generalizing Fibonacci numbers and going from there. But there are other directions that are possibly equally reasonable to study: one could look at other linear recurrences and their corresponding zeta functions. (There is some literature on this).

It is also possible to think of the Fibonacci zeta function as coming from a Pell equation, and to look at other Pell equation zeta functions. These would also have modular interpretations.

Let's say this more clearly. Fix an integer $h$ and consider the Pell equation \begin{equation} X^2 - q Y^2 = h. \end{equation} This has solutions if and only if the convolution Dirichlet series \begin{equation} \sum_{m \geq 1} \frac{\tau(m)\tau(qm + h)}{m^s} \end{equation} is nonvanishing (and this looks similar to $D_h(s)$ above, now using all $h$ and not just $h = \pm 1$).

This series can be constructed from modular forms in an analogous way to what we've done in our work.

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