title: Zeta function (operator)
text: # Zeta function (operator)
  
  The zeta function of a mathematical operator ${\mathcal {O}}$ is a function defined as
  
      $\zeta _{\mathcal {O}}(s)=\operatorname {tr} \;{\mathcal {O}}^{-s}$
  
  for those values of s where this expression exists, and as an analytic continuation of this function for other values of s. Here "tr" denotes a functional trace.
  
  The zeta function may also be expressible as a spectral zeta function in terms of the eigenvalues $\lambda _{i}$ of the operator ${\mathcal {O}}$ by
  
      $\zeta _{\mathcal {O}}(s)=\sum _{i}\lambda _{i}^{-s}$.
  
  It is used in giving a rigorous definition to the functional determinant of an operator, which is given by
  
      $\det {\mathcal {O}}:=e^{-\zeta '_{\mathcal {O}}(0)}\;.$
  
  The Minakshisundaram–Pleijel zeta function is an example, when the operator is the Laplacian of a compact Riemannian manifold.
  
  One of the most important motivations for Arakelov theory is the zeta functions for operators with the method of heat kernels generalized algebro-geometrically.
  
  