Relative class numbers and Euler–Kronecker constants of maximal real cyclotomic subfields

The Euler–Kronecker constant of a number field K K is the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function ζ K ( s ) ζ _K(s) at s = 1 s=1 . We study the distribution of the Euler–Kronecker constant γ q + γ _q^+ of the maximal real subfield of Q ( ξ q ) Q(ξ _q) , where ξ q ξ _q is the q q -th root of unity, as q q ranges over the primes. Further, we consider the distribution of γ q + − γ q γ _q^+-γ _q , with γ q γ _q the Euler–Kronecker constant of Q ( ξ q ) Q(ξ _q) and show how it is connected with Kummer’s conjecture, which predicts the asymptotic growth of the relative class number of Q ( ξ q ) Q(ξ _q) . We improve, for example, the known results on the bounds on average for the Kummer ratio and we prove analogous sharp bounds for γ q + − γ q γ _q^+-γ _q . The methods employed are partly inspired by those used by Granville [Invent. Math. 100 (1990), pp. 321–338] and Croot and Granville [J. London Math. Soc. (2) 66 (2002), pp. 579–591] to investigate Kummer’s conjecture. We supplement our theoretical findings with numerical illustrations to reinforce our conclusions.