Selberg Zeta Function

• The Selberg zeta function is defined via an Euler product over primitive hyperbolic conjugacy classes, capturing the geodesic length spectrum of hyperbolic surfaces.
• It is analytically continued using methods like the Selberg trace formula and transfer operators, linking geometric properties to Laplacian eigenvalues.
• The function plays a key role in spectral theory, quantum chaos, and arithmetic applications, influencing studies from closed geodesic distribution to black hole partition functions.

The Selberg zeta function is a central analytic object in the spectral theory of automorphic forms, quantum chaos, and the dynamics of flows on Riemannian locally symmetric spaces, most prominently finite-area hyperbolic surfaces and their higher-rank analogues. Defined both as an infinite Euler product and as a Fredholm determinant of transfer operators, it encodes the length spectrum of primitive closed geodesics and is intimately connected to the spectrum and resonances of the Laplace-Beltrami operator. Its generalizations span from arithmetic quotients to quantum field theoretic partition functions in black hole spacetimes and even to zeta functions on non-hyperbolic or p-adic and combinatorial analogues.

1. Definition and Euler Product Structure

Let $X = \Gamma \backslash \mathbb{H}$ be a (cofinite) hyperbolic surface, where $\Gamma \subset \mathrm{PSL}(2,\mathbb{R})$ is a discrete group. The Selberg zeta function is defined, for $\Re(s) > 1$, by the absolutely convergent Euler product

$$Z_\Gamma(s) = \prod_{p} \prod_{k=0}^\infty \left( 1 - N(p)^{-s-k} \right),$$

where $p$ runs over primitive hyperbolic conjugacy classes in $\Gamma$, and $N(p) = e^{\ell(p)}$ is the norm associated to the class, with $\ell(p)$ the length of the corresponding primitive closed geodesic on $X$ (Momeni et al., 2010, Kaneko et al., 2018, Garunkštis et al., 10 Nov 2025). For higher rank or orbifold quotients, appropriate modifications and twists apply: e.g., in the twisted case, the local factor at $p$ is replaced by the determinant over a representation $\Gamma \subset \mathrm{PSL}(2,\mathbb{R})$0 as in

$\Gamma \subset \mathrm{PSL}(2,\mathbb{R})$1

(Jorgenson et al., 18 Dec 2025, Momeni et al., 2011).

For infinite-area and non-compact higher-rank locally symmetric spaces, analogous Euler products are defined over suitable "primitive" classes reflecting the geometric and group-theoretic structure (Gon, 2012, Pohl, 2014).

2. Analytic Continuation, Functional Equation, and Zeros

The Euler product defining $\Gamma \subset \mathrm{PSL}(2,\mathbb{R})$2 converges absolutely for $\Gamma \subset \mathrm{PSL}(2,\mathbb{R})$3, but analytic continuation to all $\Gamma \subset \mathrm{PSL}(2,\mathbb{R})$4 is achieved via Selberg’s trace formula or via transfer operator techniques (Momeni et al., 2010, Pohl, 2014). The function becomes meromorphic or entire depending on the context (entire for convex-cocompact or compact quotients, possibly with poles at special points for non-compact finite-area cases due to continuous spectrum contributions).

A functional equation of the form

$\Gamma \subset \mathrm{PSL}(2,\mathbb{R})$5

is satisfied, where $\Gamma \subset \mathrm{PSL}(2,\mathbb{R})$6 is an explicit function built from gamma and sine factors reflecting the underlying geometry and representation data. In many cases, $\Gamma \subset \mathrm{PSL}(2,\mathbb{R})$7 reduces to elementary products in the presence of special automorphic or motivic symmetries (Koyama et al., 2020, Koyama et al., 2020).

Zeros of $\Gamma \subset \mathrm{PSL}(2,\mathbb{R})$8 encode the eigenvalues and resonances of the Laplace-Beltrami operator:

• Non-trivial zeros correspond to non-zero Laplacian eigenvalues ($\Gamma \subset \mathrm{PSL}(2,\mathbb{R})$9 if $\Re(s) > 1$0).
• Zeros lie symmetrically with respect to the critical line $\Re(s) > 1$1 (Momeni et al., 2010), with further symmetry in twisted or motivic settings (Koyama et al., 2020, Koyama et al., 2020).
• For infinite-area convex-cocompact surfaces, the rightmost zero is the Hausdorff dimension $\Re(s) > 1$2 of the limit set, and zeros organize into almost periodic patterns in the critical strip (Pollicott et al., 2022).

3. Dynamical and Operator-Theoretic Realization

A fundamental advance is the operator-theoretic realization of $\Re(s) > 1$3 as a Fredholm determinant of a transfer operator acting on spaces of holomorphic functions (Momeni et al., 2010): $\Re(s) > 1$4 where $\Re(s) > 1$5 is a suitable transfer operator associated with the geodesic flow, such as Mayer’s operator for $\Re(s) > 1$6,

$\Re(s) > 1$7

For infinite-area Hecke triangle surfaces and other situations, transfer operators are built from symbolic coding of appropriate sections of the geodesic or billiard flows (Pohl, 2014), with spectral analysis yielding determinant formulae: $\Re(s) > 1$8 where $\Re(s) > 1$9 encode boundary/weight conditions such as Dirichlet and Neumann data.

Such representations facilitate the meromorphic continuation of $$Z_\Gamma(s) = \prod_{p} \prod_{k=0}^\infty \left( 1 - N(p)^{-s-k} \right),$$0, provide explicit spectral information, and enable fast numerical calculations of zeros and resonances (Borthwick et al., 2014).

4. Spectral and Arithmetic Consequences

The Selberg zeta function is a generating function for the length spectrum of closed geodesics, and its structure mirrors the Prime Number Theorem ("prime geodesic theorem"): $$Z_\Gamma(s) = \prod_{p} \prod_{k=0}^\infty \left( 1 - N(p)^{-s-k} \right),$$1 for some $$Z_\Gamma(s) = \prod_{p} \prod_{k=0}^\infty \left( 1 - N(p)^{-s-k} \right),$$2 (Garunkštis et al., 10 Nov 2025). The locations of zeros determine asymptotics for the distribution of closed geodesics, and their relation with Laplacian eigenvalues gives rise to the Selberg trace formula.

In arithmetic settings, the zeta function connects to Dirichlet series with positive coefficients, Beurling zeta functions, and satisfies moment results: for example, the second moment at $$Z_\Gamma(s) = \prod_{p} \prod_{k=0}^\infty \left( 1 - N(p)^{-s-k} \right),$$3 exists,

$$Z_\Gamma(s) = \prod_{p} \prod_{k=0}^\infty \left( 1 - N(p)^{-s-k} \right),$$4

(Garunkštis et al., 10 Nov 2025). Generalized versions interpolate between classical Archimedean and non-Archimedean cases, with strong links to the Ihara zeta function and graph-theoretic analogues under degenerations of Schottky groups (Li et al., 2024).

In quantum chaos and mathematical physics, the Selberg zeta encapsulates quantum resonances: its zeros capture the spectrum of Laplacians on quotient manifolds, and a "prime geodesic theorem" analogy underlies analyses of quantum-classical spectral correspondences (Momeni et al., 2010, Borthwick et al., 2014).

5. Functional Equations, Motivic Twists, and Factorizations

Functional equations for generalizations of the Selberg zeta function reflect motivic automorphy structures. Twists by Tate motives via appropriate polynomial weightings yield

$$Z_\Gamma(s) = \prod_{p} \prod_{k=0}^\infty \left( 1 - N(p)^{-s-k} \right),$$5

and such twists can produce functional equations free of archimedean gamma factors if the polynomial $$Z_\Gamma(s) = \prod_{p} \prod_{k=0}^\infty \left( 1 - N(p)^{-s-k} \right),$$6 is suitably automorphic: $$Z_\Gamma(s) = \prod_{p} \prod_{k=0}^\infty \left( 1 - N(p)^{-s-k} \right),$$7 with $$Z_\Gamma(s) = \prod_{p} \prod_{k=0}^\infty \left( 1 - N(p)^{-s-k} \right),$$8 built from products over sine/gamma factors (Koyama et al., 2020, Koyama et al., 2020). This yields exact spectral symmetries not present in the untwisted case.

Symmetry reduction techniques for holomorphic iterated function schemes factorize the Selberg zeta into contributions from irreducible representations of a finite symmetry group, leading to highly efficient computation of resonances and revealing representation-theoretic structure in zero patterns (Borthwick et al., 2014).

6. Connections to Determinants, Quantum Field Theory, and Probability

There is a deep link between $$Z_\Gamma(s) = \prod_{p} \prod_{k=0}^\infty \left( 1 - N(p)^{-s-k} \right),$$9 and regularized determinants of shifted Laplacians: $p$0 (Momeni et al., 2011, Jorgenson et al., 18 Dec 2025). This connection underpins applications in quantum field theory on hyperbolic spaces, e.g., the exact 1-loop determinant on thermal AdS and BTZ black hole quotients factorizes into Selberg zeta functions evaluated at the conformal or spectral parameters of the corresponding field: $p$1 (Keeler et al., 2019, Martin et al., 2019).

Probabilistic interpretations have also emerged: the Selberg zeta evaluated at specific parameters encodes the partition function of Brownian loops on hyperbolic surfaces with killing rate. Explicitly, the total mass of essential Brownian loops is related to $p$2, allowing for probabilistic interpretations of regularized Laplacian determinants in both compact and infinite-area settings (Lemonde et al., 19 Jan 2026).

7. Extensions, Degenerate and Generalized Settings

Beyond standard hyperbolic surfaces, Selberg-type zeta functions are established for Hilbert modular surfaces, higher-rank locally symmetric spaces, and even for locally symmetric orbifolds with elliptic points, with the structure of their zeros and poles reflecting the more complex spectrum of the Laplacian and the additional geometric data (Gon, 2012).

Further, generalizations to non-hyperbolic settings, (e.g., flat space cosmologies and warped AdS black hole backgrounds) have been constructed, where the zeta function is defined through the representation theory of the global isometry group and captures spectral, quasinormal, or partition function data for quantum fields. These provide efficient, heat-kernel-free methods for computing 1-loop determinants and spectra in Lorentzian and more exotic quotient geometries (Bagchi et al., 2023, Martin et al., 2022).

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References

• (Momeni et al., 2010) Mayer Transfer Operator Approach to Selberg Zeta Function
• (Garunkštis et al., 10 Nov 2025) Selberg Zeta Functions Have Second Moment At $p$3
• (Pohl, 2014) A thermodynamic formalism approach to the Selberg zeta function for Hecke triangle surfaces of infinite area
• (Kaneko et al., 2018) Euler products of Selberg zeta functions in the critical strip
• (Koyama et al., 2020, Koyama et al., 2020) Simple/Functional equations for (generalized) Selberg zeta functions with Tate motives
• (Gon, 2012) Differences of the Selberg trace formula and Selberg type zeta functions for Hilbert modular surfaces
• (Li et al., 2024) Selberg, Ihara and Berkovich
• (Momeni et al., 2011) Zeta functions and regularized determinants related to the Selberg trace formula
• (Jorgenson et al., 18 Dec 2025) Determinants of twisted Laplacians and the twisted Selberg zeta function
• (Borthwick et al., 2014) Symmetry reduction of holomorphic iterated function schemes and factorization of Selberg zeta functions
• (Martin et al., 2019) Normal modes in thermal AdS via the Selberg zeta function
• (Keeler et al., 2019) BTZ one-loop determinants via the Selberg zeta function for general spin
• (Pollicott et al., 2022) Zeros of the Selberg zeta function for symmetric infinite area hyperbolic surfaces
• (Bagchi et al., 2023) A generalized Selberg zeta function for flat space cosmologies
• (Martin et al., 2022) A Selberg zeta function for warped AdS$p$4 black holes
• (Lemonde et al., 19 Jan 2026) Brownian Loops and the Selberg Zeta Function
• (Booker et al., 2017) Turing's method for the Selberg zeta-function