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

For a number field K, the associated Dedekind zeta function $$\zeta _K(s)$$ has a simple pole at $$s=1$$ , and we denote its residue by $$R_K$$ . Ihara introduced the Euler–Kronecker constant $$\gamma _K$$ . Let $$\ell $$ be an odd prime. We establish lower and upper bounds for $$R_K$$ and $$\gamma _K$$ when K is a cyclic extension of degree $$\ell $$ over $$\mathbb {Q}$$ . 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 $$\ell $$ .