On the Shintani zeta function for the space of binary quadratic forms
The purpose of this paper is to define an adjusted zeta function \(Z_{ad}(\Phi,\omega,s)\) of the space of quadratic forms and calculate the principal part of the poles. Shintani's zeta function is a Dirichlet series whose coefficients are the number of equivalence classes of integral quadratic forms. We can consider the adelization of Shintani's zeta function. It is denoted by \(Z(\Phi,\omega,s)\). It is a distribution with a meromorphic parameter \(s\) on the adelic space of quadratic forms. Here, \(\Phi\) is a test function of the distribution and \((\omega,s)\) are parameters of the zeta function. By reducing it to the adjusted one, \(Z_{ad}(\Phi,\omega,s)\), it becomes a meromorphic function with finite poles. The coefficients of the principal parts of the poles are also distributions. The author of this paper gives an explicit form of the principal part of \(Z_{ad}(\Phi,\omega,s)\).
scientific article; zbMATH DE number 56287
scientific article