Formal differential operators, vertex operator algebras and zeta--values, I

We study relationships between spinor representations of certain Lie algebras and Lie superalgebras of differential operators on the circle and values of $\zeta$--functions at the negative integers. By using formal calculus techniques we discuss the appearance of values of $\zeta$--functions at the negative integers underlying the construction. In addition we provide a conceptual explanation of this phenomena through several different notions of normal ordering via vertex operator algebra theory. We also derive a general Jacobi--type identity generalizing our previous construction. At the end we discuss related constructions associated to Dirichlet $L$--functions.