A residue scalar product for algebraic function fields over a number field

In 1952 Peter Roquette gave an arithmetic proof of the Riemann hypothesis for algebraic function fields of a finite constants field, which was proved by Andr'e Weil in 1940. The construction of Weil’s scalar product is essential in Roquette’s proof. In this paper a scalar product for algebraic function fields over a number field is constructed which is the analogue of Weil’s scalar product.