Values of dirichlet series at integers in the critical strip

Abstract

Oto Shimura [9], who derived them for the Dirichlet series Z m(n)n -s n:l associated to the cusp form A(z) of weight 12 for the full modular group. Somewhat later, Manin [4] extended these results to cusp forms of arbitrary integral weight for the full modular group. In the interim, Birch had introduced the ~'modular for cusp forms of weight two on F (N) and these were studied and used by Manin o [2] and [3], Mazur and Swinnerton-Dyer [5] and others. Recently, V. Miller in his thesis [6] extended the definition of the modular symbol to Fo(N). Just this year (1976), Shimura [ii], using totally different methods, has extended almost everything to F(N) and has obtained rationality results similar to those describe l below. The main result of the present note is Theorem 4. The proof consists of two main parts. The first is based on Shimura's isomorphism between cusp forms and Eichler cohomology with real coefficients. In this respect it is similar to the techniques of Manin [4]. The second part is the interpretation of the coefficients of an Eichler cocycle as residues at poles of the Mellin transform of a multiple integral of the corresponding cusp form. This is based on the Hecke correspondence (Preposition i) and on an additive character analog of Weil's theorem (Proposition 2). Well has also developed such a procedure recently in a paper delivered at the Takagi conference (1976).