Euler-Kronecker constants of maximal real cyclotomic subfields and Kummer’s conjecture

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 at $s = 1$. We study the distribution of the Euler–Kronecker constant $\gamma_q^+$ of the maximal real subfield $\mathbb{Q}(\zeta_q)^+$ 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, which predicts the asymptotic growth of the relative class number of $\mathbb{Q}(\zeta_q)$. We improve, for example, the known results on the bounds on average for the Kummer ratio and we prove analogous sharp bounds for $\gamma_q^+ - \gamma_q$. The methods employed are partly inspired by those used by Granville (1990) and Croot and Granville (2002) to investigate Kummer’s conjecture, that predicts the asymptotic growth of the relative class number of prime cyclotomic fields. We substantially improve the known bounds of Kummer’s ratio under three scenarios: no Siegel zero, presence of Siegel zero and assuming the Riemann Hypothesis for the Dirichlet $L$-series attached to odd characters only. The talk is based on joint papers with A. Languasco, P. Moree, N. Kandhil and S. Saad Eddin.

number theory

Audience: researchers in the topic

Combinatorial and additive number theory seminar (CANT 2026)