# An Introduction to Ihara Zeta Function
## Zero and Pole
A $\textbf{function of a complex variable}$ $z$ is $\textbf{holomorphic}$ in an $\textbf{open domain}$ $U$ if it is $\textbf{differentiable}$ with respect to $z$ at every point of $U$. Equivalently, it is holomorphic if it is $\textbf{analytic}$, that is, if its $\textbf{Taylor series}$ exists at every point of $U$, and converges to the function in some $\textbf{neighbourhood}$ of the point. A function is $\textbf{meromorphic}$ in $U$ if every point of $U$ has a neighbourhood such that at least one of $f$ and $1/f$ is holomorphic in it.

A $\textbf{zero}$ of a meromorphic function $f$ is a complex number $z$ such that $f(z) = 0$. A $\textbf{pole}$ of $f$ is a zero of $1/f$.
## A Quick Introduction to Riemann Zeta Function and Riemann Hypothesis

$\textbf{The Riemann zeta function}$ is defined for $Re(s) > 1$ by
\begin{equation}
\zeta(s) = \sum_{n=1}^{\infty} \frac{1}{n^s} = \prod_{p=\text{prime}} \left(1 - p^{-s}\right)^{-1}.
\end{equation}

In 1859 Riemann extended this function to all complex numbers $s$ as a meromorphic function with its only pole (a simple one) at $s = 1$. He also showed that there is a functional equation relating $\zeta(s)$ and $\zeta(1-s)$. The functional equation involves the gamma function and says
\begin{equation}
\Lambda(s) = \pi^{-s/2} \Gamma \left( \frac{s}{2} \right) \zeta(s) = \Lambda(1-s).
\end{equation}

Riemann's main interest in zeta was as a means to prove $\textbf{the prime number theorem}$ (proved by Hadamard and de la Vallée Poussin in 1896) which says
\begin{equation}
\# \{ p \mid p = \text{prime}, p \leq x \} \sim \frac{x}{\ln x}, \quad \text{as} \quad x \to \infty.
\end{equation}The Hadamard product for $s(s-1)\Lambda(s)$ gives a relation between the primes and the zeros of zeta. There are always zeros at the negative even integers (the trivial zeros). The still unproved Riemann hypothesis says the non-trivial zeros of $\zeta(s)$ are on the line $Re(s) = \frac{1}{2}$. It has been checked for a ridiculously large number of zeros. The Riemann hypothesis has implications for the error term in the prime number theorem.

## Ihara Zeta Function

The graph theory zetas first appeared in work of Ihara on $p$-adic groups in the 1960s. Soon the connection with graphs was found and many papers appeared. The main authors in the 1980s and 90s were Sunada, Hashimoto , and Bass . Other references are Venkov and Nikitin and Northshield's paper in the volume of Hejhal . The main properties of the Riemann zeta function have graph theory analogs, at least for regular graphs. For irregular graphs, there is no known functional equation and it is difficult to formulate the Riemann hypothesis.

We will always assume that our graphs $X$ are finite, connected, rank $\geq 1$ with no danglers (i.e., degree 1 vertices). Here "rank" means the rank of the fundamental group of the graph . Let us recall some of the definitions basic to Stark and Terras.

If $X$ is any connected finite undirected graph with vertex set $V$ and (undirected) edge set $E$, we orient its edges arbitrarily and obtain $2|E|$ oriented edges labelled
\begin{equation}
e_1, e_2, \cdots, e_n, e_{n+1} = e_1^{-1}, \ldots, e_{2n} = e_n^{-1}.
\end{equation}

"Primes" $[P]$ in $X$ are equivalence classes of closed backtrackless tailless primitive paths $P$. Write a path $C = a_1 a_2 \cdots a_s$, where $a_j$ is an oriented edge of $X$. The length of $C$ is $\nu(C) = s$. $\textbf{Backtrackless}$ means that $a_{i+1} \neq a_i^{-1}$ for all $i$. $\textbf{Tailless}$ means that $a_s \neq a_1^{-1}$. The $\textbf{equivalence class}$ $[C]$ is the set

\begin{equation}
[C] = \{a_1 a_2 \cdots a_s, a_2 a_3 \cdots a_1, \ldots, a_s a_1 \cdots a_{s-1}\}.
\end{equation}

$[P]$ is $\textbf{primitive}$ means $P \neq D^m$, for any integer $m \geq 2$ and path $D$ in $X$. Here $r$ will denote the rank of the fundamental group of $X$. We have $r - 1 = |E| - |V|$. Then $r$ is the number of edges deleted from $X$ to form a spanning tree.

The $\textbf{Ihara (vertex) zeta function}$ of $X$ is defined at $u \in \mathbb{C}$, for $|u|$ sufficiently small, by
\begin{equation}
\zeta_X(u) = \zeta(u, X) = \prod_{[P]} \left(1 - u^{\nu(P)}\right)^{-1},
\end{equation}where $[P]$ runs over the primes of $X$. Here $V$ is for vertex. In this section, we will drop $V$ most of the time for the vertex zeta. In later sections we will define two other kinds of zeta associated to a graph.

$\textbf{Definition}$
$R_X$ is the $\textbf{radius of the largest circle of convergence}$ of the Ihara zeta function.
When $X$ is a $(q + 1)$-regular graph, $R_X = 1/q$.

The generalization of $\textbf{Ihara's formula}$ (proved by Hashimoto and Bass) is
\begin{equation}
\zeta_X(u)^{-1} = (1 - u^2)^{r-1} \det(I - Au + Qu^2).
\end{equation}

Suppose that $X$ is a $(q + 1)$-regular graph. Then the Ihara zeta function has functional equations relating the value at $u$ to the value at $1/(qu)$. Setting $u = q^{-s}$, one finds that there is a functional equation relating the value at $s$ to that at $1 - s$, just as for Riemann’s zeta function.(see below)
$\textbf{Proposition}$
Suppose that $X$ is a $(q + 1)$-regular connected graph without degree-1 vertices and with $n = |V|$. Then we have the following functional equations, among others.

(1) $\Lambda_X(u) \equiv (1 - u^2)^{r-1+n/2}(1 - q^2 u^2)^{n/2} \zeta_X(u) = (-1)^n \Lambda_X(1/qu)$.

(2) $\xi_X(u) \equiv (1 + u)^{r-1}(1 - u)^{r-1+n}(1 - qu)^n \zeta_X(u) = \xi_X(1/qu)$.

(3) $\mathfrak{X}_X(u) \equiv (1 - u^2)^{r-1}(1 + qu)^n \zeta_X(u) = \mathfrak{X}_X(1/qu)$.

As with the Dedekind zeta function, $\zeta_X(u)$ has a meromorphic continuation to the entire complex $u$-plane, but now $\zeta_X(u)^{-1}$ is entire. Thus if we want to formulate a version of the Riemann Hypothesis, we must consider the poles of $\zeta_X(u)$, not the zeroes.

$\textbf{Example}$
Consider the tetrahedron graph $K_4$ has Ihara zeta function
\begin{equation}
\zeta_{K_4}(u)^{-1} = (1 - u^2)^2 (1 - u)(1 - 2u)(1 + u + 2u^2)^3.
\end{equation}

We first draw a contour map of $z = |\zeta_{K_4}(x + iy)|^{-1}$,but it is easy to see the contour map of $z = |\zeta_{K_4}(2^{-(x+iy)})|^{-1}$ is more like that for the Riemann zeta function.

$\textbf{Figure}$ A contour map of the Ihara zeta for $K_4$ in the $u$-variable: $z = 1/|\zeta_{K_4}(x + iy)|$.

$\textbf{Figure}$ (left)A contour map of zeta for $K_4$ in the $s$-variable: $z = 1/|\zeta_{K_4}(2^{-(x+iy)})|$.(right) Graph of the Riemann zeta, i.e., $z = |\zeta(x + iy)|$

## Riemann Hypothesis of Regular graph

$\textbf{Definition}$
A connected $(q + 1)$-regular graph $X$ is $\textit{Ramanujan}$ if and only if, for
\begin{equation}
\mu = \max \{ |\lambda| \mid \lambda \in \text{Spectrum } A, \ |\lambda| \neq q + 1 \},
\end{equation}we have $\mu \leq 2\sqrt{q}$.

Some graphs are Ramanujan and some are not. In the 1980s, Margulies and Lubotzky found a construction of infinite families of Ramanujan graphs of fixed degree equal to $1 + p^e$, where $p$ is a prime. They used the Ramanujan conjecture (now proved by Deligne) to show that the graphs were Ramanujan. Such graphs are of interest to computer scientists because they provide efficient communication as they have good expansion properties.Friedman proved that a random regular graph is almost Ramanujan (the Alon conjecture).

For experiments leading to the conjecture that the percentage of regular graphs exactly satisfying the RH approaches 27\% as the number of vertices approaches infinity. The argument involves the Tracy–Widom distribution from random matrix theory.

Now let's see the Riemann hypothesis of regular graph
$\textbf{Definition}$
Suppose that $X$ is a connected $(q + 1)$-regular graph (without degree-1 vertices). We say that the Ihara zeta function $\zeta_X(q^{-s})$ satisfies the $\textbf{Riemann hypothesis}$ iff, when $0 < Re(s) < 1$,
\begin{equation}
\zeta_X(q^{-s})^{-1} = 0 \implies Re(s) = \frac{1}{2}.
\end{equation}

Note that, if $u = q^{-s}$, $Re(s) = \frac{1}{2}$ corresponds to $|u| = 1/\sqrt{q}$.

then we can see our main reslut

$\textbf{Theorem}$
For a connected $(q + 1)$-regular graph $X$, $\zeta_X(u)$ satisfies the $\textbf{Riemann hypothesis}$ iff the graph $X$ is Ramanujan in the sense of Definition above.

In order to proof this theorem,we fist see some property about spectrum of A the adjacency matrix

$\textbf{Proposition}$
(Facts about Spectrum A, when A is the adjacency operator of a connected $(q + 1)$-regular graph $X$) Assume that $X$ is a connected $(q + 1)$-regular graph and that $A$ is its adjacency matrix. Then:
(1). $\lambda \in \text{Spectrum } A \implies |\lambda| \leq q + 1$;

(2). $q + 1 \in \text{Spectrum } A \text{ and has multiplicity } 1$;

$\textit{Proof of fact (1)}$
Note that $q + 1$ is clearly an eigenvalue of $A$ corresponding to the constant vector. Suppose that $Av = \lambda v$, for some column vector $v = \text{t}(v_1 \cdots v_n) \in \mathbb{R}^n$, and suppose that the maximum $|v_i|$ occurs at $i = a$. Then, using the notation $b \sim a$ to mean that the $b$th vertex is adjacent to the $a$th, we have
\begin{equation}
|\lambda||v_a| = |(Av)_a| = \left|\sum_{b \sim a} v_b\right| \leq (q + 1)|v_a|.
\end{equation}

Fact (1) follows.

$\textit{Proof of fact (2)}$
Suppose that $Av = (q + 1)v$ for some non-zero vector $v = \text{t}(v_1, \ldots, v_n) \in \mathbb{R}^n$. Again suppose that the maximum $|v_i|$ occurs at $i = a$. We can assume $v_a > 0$, since we can multiply the vector $v$ by $-1$ if necessary. As in the proof of fact (1),
\begin{equation}
(q + 1)v_a = (Av)_a
\end{equation}
\begin{equation}
= \sum_{b \sim a} v_b \leq (q + 1)v_a.
\end{equation}

To have equality, there can be no cancellation in this sum and therefore $v_b = v_a$ for each $b$ adjacent to $a$. Since we have assumed that $X$ is connected, we can iterate this argument and conclude that $v$ must be the constant vector.

$\textbf{Corollary}$

$-(q + 1) \in \text{Spectrum } A \text{ iff the graph } X \text{ is bipartite}$.

Then we can proof our main result

$\textit{Proof}$
Use Ihara determinant formula
\begin{equation}
\zeta_X(q^{-s})^{-1} = (1 - u^2)^{r-1} \prod_{\lambda \in \text{Spectrum } A} (1 - \lambda u + q u^2).
\end{equation}Write $1 - \lambda u + q u^2 = (1 - \alpha u)(1 - \beta u)$, where $\alpha \beta = q$ and $\alpha + \beta = \lambda$. Note that $\alpha, \beta$ are the reciprocals of poles of $\zeta_X(u)$. Using the facts in Proposition above, we have three cases.

$\textit{Case 1}$ $\quad\lambda = \pm (q + 1)$, implying that $\alpha = \pm q$ and $\beta = \pm 1$.

$\textit{Case 2}$ $\quad|\lambda| \leq 2\sqrt{q}$, implying that $|\alpha| = |\beta| = \sqrt{q}$.

$\textit{Case 3}$ $\quad2\sqrt{q} < |\lambda| < q + 1$, implying that $\alpha, \beta \in \mathbb{R}$ and $1 < |\alpha| = |\beta| < q$, $|\alpha| = |\beta| \neq \sqrt{q}$.

To deduce these results, let $u$ be either $\alpha^{-1}$ or $\beta^{-1}$. Then, by the quadratic formula, we have $\alpha$ or $\beta = u^{-1}$ where
\begin{equation}
u = \left( \lambda \pm \sqrt{\lambda^2 - 4q} \right) / (2q).
\end{equation}

The pole of Cases 1 is simply four points and 2 are circle centered 0 with radius $1/\sqrt{q}$. To understand case 3, first assume that $\lambda > 0$ and note that $u = (\lambda + \sqrt{\lambda^2 - 4q}) / (2q)$ is a monotone increasing function of $\lambda$. This implies that the larger root $u$ is in the interval $(1/\sqrt{q}, 1)$. Where does the smaller root $u' = (\lambda - \sqrt{\lambda^2 - 4q}) / (2q)$ lie? Answer: $|u'| \in (1/q, 1/\sqrt{q})$. Here we have used the fact that $uu' = 1/q$. A similar argument works for negative $\lambda$. The proof of the theorem is concluded by noting that when $u = q^{-s}$, case 2 gives $Re(s) = 1/2$.

$\textbf{Figure }$ The possible locations for the poles of $\zeta_X(u)$ for a regular graph are the circle and the real-axis sections shown. The circle corresponds to the part of the spectrum of the adjacency matrix satisfying the Ramanujan inequality. The real poles correspond to the non-Ramanujan eigenvalues of $A$, except for the poles on the circle itself and those at the endpoints of the intervals.

The above figure shows the possible locations of the poles of the Ihara zeta function of a $(q + 1)$-regular graph. The poles satisfying the Riemann hypothesis are those on the circle; the circle basically corresponds to case 2 in the preceding proof. The real axis sections correspond to cases 1 and 3.This is why the radius of convergence of the product defining the Ihara zeta function of a $(q + 1)$-regular graph is $R_X = 1/q$.

We have seem their are some functional equations for the Ihara zeta function corresponding to a regular graph. If we set $u = q^{-s}$, the functional equations relate the value at $s$ to that at $1 - s$, just as in the case of the Riemann zeta function.

Look at figure. What sort of symmetry is indicated by the functional equations, which imply that if $\zeta_X(u)$ has a pole at $u$ then it must also have a pole at $1/(qu)$? If $u$ is on the circle then $1/(qu)$ is the complex conjugate of $u$. If $u$ is in the interval $(1/\sqrt{q}, 1)$ then $1/(qu)$ is in the interval $(1/q, 1/\sqrt{q})$.

$\textbf{Example}$
The tetrahedron graph $K_4$ in Figure 2.5 has Ihara zeta function
\begin{equation}
\zeta_{K_4}(u)^{-1} = (1 - u^2)^2 (1 - u)(1 - 2u)(1 + u + 2u^2)^3.
\end{equation}

The five poles of this zeta function are located at the points $-1, 1/2, 1, (-1 \pm \sqrt{-7})/4$. The absolute value of the complex pole is $1/\sqrt{2} \approx 0.70711$. The closest pole to the origin is $1/2 = 1/q = R_{K_4}$. Of course, $K_4$ is a Ramanujan graph and thus the Riemann hypothesis holds for this graph.

## Graph theory prime number theorem
One can use the Ihara zeta function to prove the graph prime number theorem. In order to state this result, we need some definitions.

$\textbf{Definition}$
The prime counting function is
\begin{equation}
\pi(n) = \#\{\text{primes } [P] \mid n = \nu(P) = \text{length of } P\}.
\end{equation}You may find it a little surprising to note that we have replaced \(\leq\) in the usual prime counting function of formula (1.2) with an equals sign here.

$\textbf{Definition}$
The greatest common divisor of the prime path lengths is
\begin{equation}
\Delta_X = \text{g.c.d.} \left\{ \nu(P) \mid [P] \text{ prime of } X \right\}.
\end{equation}

$\textbf{The graph theory prime number theorem}$ for a graph $X$ satisfying the usual hypotheses, stated at the beginning of Section 2.1, says that if $\Delta_X$ divides $m$ then
\begin{equation}
\pi(m) \sim \frac{\Delta_X}{mR_X^m} \text{ as } m \rightarrow \infty. \tag{2.4}
\end{equation}

If $\Delta_X$ does not divide $m$ then $\pi(m) = 0$. Note that the theorem is clearly false for the bad graph sush as below


## Ihara Zeta Function of a Weighted Graph

Many applications involve weighted or metric graphs, that is, graphs with positive real numbers attached to the edges to represent length or resistance or some other physical attribute. In particular, quantum graphs are weighted.For the most part, we will not consider weighted graphs here but let us at least give a natural extension of the definition of the Ihara zeta function to weighted graphs.

$\textbf{Definition}$
For a graph $X$ with oriented edge set $\overrightarrow{E}$, consisting of $2|E|$ oriented edges, suppose that we have a weighting function $L : \overrightarrow{E} \rightarrow \mathbb{R}^+$. Then define the $\textbf{weighted length}$ of a closed path $C = a_1 a_2 \cdots a_s$, where $a_j \in \overrightarrow{E}$, by
\begin{equation}
\nu(C, L) = \nu_X(C, L) = \sum_{i=1}^s L(a_i).
\end{equation}

$\textbf{Definition}$
The Ihara zeta function of a weighted (undirected) graph for $|u|$ small and $u \notin (-\infty, 0)$ is
\begin{equation}
\zeta_X(u, L) = \prod_{[P]} \left(1 - u^{\nu(P, L)}\right)^{-1}.
\end{equation}

Clearly, when $L = 1$, meaning the function such that $L(e) = 1$ for all edges $e$ in $X$, we have $\zeta_X(u, 1) = \zeta_X(u)$, our original Ihara zeta function.

$\textbf{Definition 6.3}$
Given a graph $X$ with positive integer-valued weight function $L$, define the $\textbf{inflated graph}$ $X_L$ in which each edge $e$ is replaced by an edge with $L(e) - 1$ new degree-2 vertices.

Then clearly $\nu_X(C, L) = \nu_{X_L}(C, 1)$, where the argument 1 means again that $L(e) = 1$ for all edges $e$. It follows that $\textbf{for positive integer-valued weights}$ $L$ we have the following identity relating the weighted zeta and the ordinary Ihara zeta:
\begin{equation}
\zeta_X(u, L) = \zeta_{X_L}(u).
\end{equation}

Therefore $\zeta_X(u, L)^{-1}$ is a polynomial for integer-valued weights $L$.

For non-integer weights, it is possible to obtain a determinant formula using the edge zeta functions we will discuss it inthe future.