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

Abstract

The Euler--Kronecker constant of a number field $K$ is the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $ζ_K(s)$ at $s=1$. We study the distribution of the Euler--Kronecker constant $γ_q^+$ of the maximal real subfield of $\mathbb Q(ζ_q)$ as $q$ ranges over the primes. Further, we consider the distribution of $γ_q^+-γ_q$, with $γ_q$ the Euler--Kronecker constant of $\mathbb Q(ζ_q)$ and show how it is connected with Kummer's conjecture, which predicts the asymptotic growth of the relative class number of $\mathbb Q(ζ_q)$. We improve, for example, the known results on the bounds on average for the Kummer ratio and we prove analogous sharp bounds for $γ_q^+-γ_q$. The methods employed are partly inspired by those used by Granville (1990) and Croot and Granville (2002) to investigate Kummer's conjecture. We supplement our theoretical findings with numerical illustrations to reinforce our conclusions.