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:

$$ \zeta _K \left( s \right) = c_{ - 1} \left( {s - 1} \right)^{ - 1} + c_0 + c_1 \left( {s - 1} \right) + \cdots \left( {c_{ - 1} \ne 0} \right) $$

(0.1)

In this paper, we shall present a systematic study of the real number

$$ \gamma _K = {{c_0 } \mathord{\left/ {\vphantom {{c_0 } {c_{ - 1} }}} \right. \kern-\nulldelimiterspace} {c_{ - 1} }} $$

(0.2)

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

$$ \gamma _\mathbb{Q} = \mathop {\lim }\limits_{n \to \infty } \left( {1 + \frac{1} {2} + \cdots + \frac{1} {n} - \log n} \right) = 0.57721566..., $$

and when K is imaginary quadratic, the well-known Kronecker limit formula expresses γ _K in terms of special values of the Dedekind η function. This constant γ _K 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 γ _K more as an invariant of K.