Irregular behaviour of class numbers and Euler-Kronecker constants of cyclotomic fields: the log log log devil at play

TL;DR

This paper surveys the irregular behavior of class numbers and Euler-Kronecker constants in cyclotomic fields, highlighting their similar unpredictable patterns influenced by prime distribution irregularities.

Contribution

It reveals the analogous irregularities in class numbers and Euler-Kronecker constants, connecting Kummer's and Ihara's conjectures through prime distribution anomalies.

Findings

Both invariants exhibit irregular behavior influenced by prime distribution.

Under standard conjectures, the conjectures hold for most primes but fail in general.

The objects studied by Kummer and Ihara show similar unpredictable patterns.

Abstract

Kummer (1851) and, many years later, Ihara (2005) both posed conjectures on invariants related to the cyclotomic field $\mathbb Q(\zeta_q)$ with $q$ a prime. Kummer's conjecture concerns the asymptotic behaviour of the first factor of the class number of $\mathbb Q(\zeta_q)$ and Ihara's the positivity of the Euler-Kronecker constant of $\mathbb Q(\zeta_q)$ (the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function $\zeta_{\mathbb Q(\zeta_q)}(s)$ at $s=1$). If certain standard conjectures in analytic number theory hold true, then one can show that both conjectures are true for a set of primes of natural density 1, but false in general. Responsible for this are irregularities in the distribution of the primes. With this survey we hope to convince the reader that the apparently dissimilar mathematical objects studied by Kummer and Ihara actually display a very similar behaviour.