Exploring the interplay of Euler-Kronecker constants, prime numbers, and Kummer's Conjecture

The Euler-Kronecker constant of a number field $K$ is defined as the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function associated with $K$ at $s=1$. Investigating the distribution of the normalized difference of the Euler-Kronecker constants of the prime cyclotomic field $\mathbb{Q}(\zeta_q)$ and its maximal real subfield, we connect it with Kummer's conjecture. This conjecture predicts the asymptotic growth of the relative class number of prime cyclotomic fields. The ratio of this class number and the simple asymptotic that Kummer predicted for it is now called the Kummer ratio. Furthermore, we discuss recent advancements in understanding the irregularities in the behavior of the Kummer ratio shedding light on deep questions related to prime number distribution.

This talk is based on several works with A. Languasco, P. Moree, S. Saad Eddin and A. Sedunova.