Relative class numbers and Euler-Kronecker constants of maximal real subfields (NOTE THE DATE!)

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 $\zeta_K(s)$ at $s=1$. We study the distribution of the Euler-Kronecker constant $\gamma_q^+$ of the maximal real subfield of $\mathbb Q(\zeta_q)$ as $q$ ranges over the primes. Further, we consider the distribution of $\gamma_q^+-\gamma_q$, with $\gamma_q$ the Euler--Kronecker constant of $\mathbb Q(\zeta_q)$ and show how it is connected with Kummer's conjecture. This conjecture predicts the asymptotic growth of the relative class number of $\mathbb Q(\zeta_q)$. We improve the known results on the bounds on average for the Kummer ratio and prove analogous sharp bounds for $\gamma_q^+ \gamma_q$.
(Joint work with A. Languasco, P. Moree, A. Sedunova, and S. Saad Eddin)