On the residues and Euler-Kronecker constants of cyclic number fields

Abstract

For a number field K, the associated Dedekind zeta function ζK (s) has a simple poleat s = 1, and we denote its residue by RK . Ihara introduced the Euler–Kronecker constant γK . Let be an odd prime. We establish lower and upper bounds for RK and γK when K is a cyclic extension of degree over Q. These bounds are stronger than those known under the Generalized Riemann Hypothesis (GRH) and are shown to be sharp. However, the trade-off is that they hold only almost surely. Finally, we compute the average of the Euler–Kronecker constants for cyclic fields K of degree .