Zeta function (operator)

The <a href="/facts/Zeta_function_(disambiguation)/lf43angF">zeta function</a> of a mathematical <a href="/facts/Operator_(mathematics)/1VWgEdjS">operator</a> 
 
 
 
 
 
 O
 
 
 
 
 {\displaystyle {\mathcal {O}}}
 
 is a function defined as

ζ
          
            
              O
            
          
        
        (
        s
        )
        =
        tr
        
        
          
            
              O
            
          
          
            −
            s
          
        
      
    
    {\displaystyle \zeta _{\mathcal {O}}(s)=\operatorname {tr} \;{\mathcal {O}}^{-s}}

for those values of s where this expression exists, and as an <a href="/facts/Analytic_continuation/aBCuEgvt">analytic continuation</a> of this function for other values of s. Here "tr" denotes a functional <a href="/facts/Trace_(linear_algebra)/dchFFvHt">trace</a>.
The zeta function may also be expressible as a spectral zeta function in terms of the <a href="/facts/Eigenvalues/8TjEoT8u">eigenvalues</a> 
 
 
 
 
 λ
 
 i
 
 
 
 
 {\displaystyle \lambda _{i}}
 
 of the operator 
 
 
 
 
 
 O
 
 
 
 
 {\displaystyle {\mathcal {O}}}
 
 by

ζ
 
 
 O
 
 
 
 (
 s
 )
 =
 
 ∑
 
 i
 
 
 
 λ
 
 i
 
 
 −
 s
 
 
 
 
 {\displaystyle \zeta _{\mathcal {O}}(s)=\sum _{i}\lambda _{i}^{-s}}
 
.
It is used in giving a rigorous definition to the <a href="/facts/Functional_determinant/preMcoEw">functional determinant</a> of an operator, which is given by

det
        
          
            O
          
        
        :=
        
          e
          
            −
            
              ζ
              
                
                  O
                
              
              ′
            
            (
            0
            )
          
        
        
        .
      
    
    {\displaystyle \det {\mathcal {O}}:=e^{-\zeta '_{\mathcal {O}}(0)}\;.}

The <a href="/facts/Minakshisundaram%E2%80%93Pleijel_zeta_function/dwThxTA9">Minakshisundaram–Pleijel zeta function</a> is an example, when the operator is the <a href="/facts/Laplace_operators_in_differential_geometry/LT7U1qjj">Laplacian</a> of a compact <a href="/facts/Riemannian_manifold/5KYnmEgF">Riemannian manifold</a>.
One of the most important motivations for <a href="/facts/Arakelov_theory/JwpLlcMw">Arakelov theory</a> is the zeta functions for operators with the method of <a href="/facts/Heat_kernel/EbLKLbJH">heat kernels</a> generalized algebro-geometrically.

Zeta function (operator) open-in-new

Zeta function (operator)