Selberg zeta function

The Selberg zeta-function was introduced by <a href="/facts/Atle_Selberg/3JDSUN2k">Atle Selberg</a> (1956). It is analogous to the famous <a href="/facts/Riemann_zeta_function/MnfRu7lY">Riemann zeta function</a>

ζ
        (
        s
        )
        =
        
          ∏
          
            p
            ∈
            
              P
            
          
        
        
          
            1
            
              1
              −
              
                p
                
                  −
                  s
                
              
            
          
        
      
    
    {\displaystyle \zeta (s)=\prod _{p\in \mathbb {P} }{\frac {1}{1-p^{-s}}}}

where 
 
 
 
 
 P
 
 
 
 {\displaystyle \mathbb {P} }
 
 is the set of prime numbers. The Selberg zeta-function uses the lengths of simple <a href="/facts/Closed_geodesic/DlBzcCgT">closed geodesics</a> instead of the prime numbers. If 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
 is a subgroup of <a href="/facts/SL2(R)/xjKM9JLJ">SL(2,R)</a>, the associated Selberg zeta function is defined as follows,

ζ
          
            Γ
          
        
        (
        s
        )
        =
        
          ∏
          
            p
          
        
        (
        1
        −
        N
        (
        p
        
          )
          
            −
            s
          
        
        
          )
          
            −
            1
          
        
        ,
      
    
    {\displaystyle \zeta _{\Gamma }(s)=\prod _{p}(1-N(p)^{-s})^{-1},}

or

Z
          
            Γ
          
        
        (
        s
        )
        =
        
          ∏
          
            p
          
        
        
          ∏
          
            n
            =
            0
          
          
            ∞
          
        
        (
        1
        −
        N
        (
        p
        
          )
          
            −
            s
            −
            n
          
        
        )
        ,
      
    
    {\displaystyle Z_{\Gamma }(s)=\prod _{p}\prod _{n=0}^{\infty }(1-N(p)^{-s-n}),}

where p runs over conjugacy classes of <a href="/facts/Prime_geodesic/sz0R6Qqm">prime geodesics</a> (equivalently, conjugacy classes of primitive hyperbolic elements of 
 
 
 
 Γ
 
 
 {\displaystyle \Gamma }
 
), and N(p) denotes 
 
 
 
 exp
 ⁡
 (
 
 length of 
 
 p
 )
 
 
 {\displaystyle \exp({\text{length of }}p)}
 
 (equivalently, the square of the bigger eigenvalue of p).
For any <a href="/facts/Hyperbolic_manifold/eUBhz9uW">hyperbolic surface</a> of finite area there is an associated Selberg zeta-function; this function is a <a href="/facts/Meromorphic_function/LeNBd6Sh">meromorphic function</a> defined in the <a href="/facts/Complex_plane/vE0QrxJp">complex plane</a>. The zeta function is defined in terms of the closed <a href="/facts/Geodesic/fbWR6l80">geodesics</a> of the surface.
The zeros and poles of the Selberg zeta-function, Z(s), can be described in terms of spectral data of the surface.
The zeros are at the following points:

<ol><li>For every cusp form with eigenvalue 
 
 
 
 
 s
 
 0
 
 
 (
 1
 −
 
 s
 
 0
 
 
 )
 
 
 {\displaystyle s_{0}(1-s_{0})}
 
 there exists a zero at the point 
 
 
 
 
 s
 
 0
 
 
 
 
 {\displaystyle s_{0}}
 
. The order of the zero equals the dimension of the corresponding eigenspace. (A cusp form is an eigenfunction to the <a href="/facts/Laplace%E2%80%93Beltrami_operator/AJmyUpWz">Laplace–Beltrami operator</a> which has <a href="/facts/Fourier_expansion/s3fsxPAT">Fourier expansion</a> with zero constant term.)</li>
<li>The zeta-function also has a zero at every pole of the determinant of the scattering matrix, 
 
 
 
 ϕ
 (
 s
 )
 
 
 {\displaystyle \phi (s)}
 
. The order of the zero equals the order of the corresponding pole of the scattering matrix.</li></ol>
The zeta-function also has poles at 
 
 
 
 1
 
 /
 
 2
 −
 
 N
 
 
 
 {\displaystyle 1/2-\mathbb {N} }
 
, and can have zeros or poles at the points 
 
 
 
 −
 
 N
 
 
 
 {\displaystyle -\mathbb {N} }
 
.
The <a href="/facts/Ihara_zeta_function/1FJ8MMr9">Ihara zeta function</a> is considered a p-adic (and a graph-theoretic) analogue of the Selberg zeta function.

Selberg zeta function open-in-new

Selberg zeta function