On the euler–kronecker constants of global fields and primes with small norms

Abstract

Let K be a global field, i.e., either an algebraic number field of finite degree (abbreviated NF), or an algebraic function field of one variable over a finite field (FF). Let ζK(s) be the Dedekind zeta function of K, with the Laurent expansion at s = 1: (formula presented) In this paper, we shall present a systematic study of the real number attached to each K, which we call the Euler-Kronecker constant (or invariant) of K. When K = Q (the rational number field), it is nothing but the Euler-Mascheroni constant (formula presented) and when K is imaginary quadratic, the well-known Kronecker limit formula expresses γK in terms of special values of the Dedekind η function. This constant γK appears here and there in several articles in analytic number theory, but as far as the author knows, it has not played a main role nor has it been systematically studied. We shall consider γK more as an invariant of K.