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

TL;DR

This paper investigates the distribution of Euler--Kronecker constants in maximal real subfields of cyclotomic fields, connecting these constants with Kummer's conjecture and providing new bounds and numerical evidence.

Contribution

It introduces new bounds on the distribution of Euler--Kronecker constants and their differences, linking them to Kummer's conjecture, with both theoretical and numerical analysis.

Findings

Established bounds on the average of the Kummer ratio

Proved sharp bounds for the difference of Euler--Kronecker constants

Numerical illustrations support the theoretical results

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 $\zeta_K(s)$ 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, 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. We supplement our theoretical findings with numerical illustrations to reinforce our conclusions.