Prime Number Theorem for Algebraic Function Fields

Elementary proofs of the abstract prime number theorem of the form A(w) = qm + 0(qmm~i) for algebraic function fields are given. The proofs use a refinement of a tauberian theorem of Bombieri. 0. Introduction The main purpose of this paper is to give elementary proofs of the abstract prime number theorem for algebraic function fields (henceforth, the P.N.T.) which was established in the author's paper [8]. In the analytic argument in [8], the fact that the generating function Z*(y) has no zeros on the circle \y\ — q~] plays an important role. In this paper, we shall show that the nonvanishing of the generating function has an elementary interpretation which can be expressed as another analogue (see (2.3)) of Selberg's formula in classical prime number theory. In a recent paper [2], Bombieri also proposed this formula. By introducing the new analogue and proving an elegant tauberian theorem (see Theorem 3.1), he made a correction to the last part of the argument in his earlier paper [1]. In this paper, we shall refine this tauberian theorem in a quantitative form. The result, Theorem 3.2, is sharp in some sense. Combining our elementary proof of the new analogue of Selberg's formula and the new tauberian theorem will give an elementary proof of the P.N.T. with a remainder term (see Theorems 4.1 and 4.2). Since the last part of the argument given in [1] is inadequate to prove the P.N.T., it is an interesting question whether the P.N.T. (or the conclusion in [1]) can be established with the aid of the new analogue mentioned above. We shall show briefly that this is indeed the case. The author thanks Professor E. Bombieri for the preprint of his article [2]. 1. An elementary proof of the first analogue of Selberg's formula We begin by recalling the concept of an additive arithmetic semigroup, which was developed by Knopfmacher [6]. An additive arithmetic semigroup is, by Received by the editors November 21, 1989 and, in revised form, June 11, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 11N80; Secondary 11T41, 11T55.