and deduce from it the explicit formula of our global zeta functions. As is explained later, the nonvanishing of infinitely many periods immediately follows from this explicit formula.

Methods and details of the results
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The main point of proving Theorem \[thm:main\] is a new way to show nonvanishing of infinitely many period integrals. The setting of our question seems purely algebraic, but it seems that we need the methods and techniques of analytic number theory. A basic idea in analytic number theory to handle such problems is relating them to the existence of poles or zeros of some Dirichlet series or other similar infinite series. One of the most typical examples is the Dirichlet prime number theorem and its generalization called the Chebotarev’s density theorem. These theorems assert the existence of infinite number of prime numbers or places of number fields with certain properties, and are proved by analyzing Dirichlet and Dirichlet-Hecke $L$-functions.

In this paper, we introduce a new method along the same lines as these previous works with a different tool, the prehomogeneous zeta functions with period integrals. This method combines the techniques of analytic number theory with analysis of prehomogeneous zeta functions *à la* H. Saito 