Irregular Behaviour of Class Numbers and Euler-Kronecker Constants of Cyclotomic Fields: The Log Log Log Devil at Play

Kummer (1851) and, many years later, Ihara (2005) both posed conjectures on invariants related to the cyclotomic field ℚ ( ζ q ) $$\mathbb Q(\zeta _q)$$ with q a prime. Kummer’s conjecture concerns the asymptotic behaviour of the first factor of the class number of ℚ ( ζ q ) $$\mathbb Q(\zeta _q)$$ and Ihara’s the positivity of the Euler-Kronecker constant of ℚ ( ζ q ) $$\mathbb Q(\zeta _q)$$ (the ratio of the constant and the residue of the Laurent series of the Dedekind zeta function ζ ℚ ( ζ q ) ( s ) $$\zeta _{\mathbb Q(\zeta _q)}(s)$$ 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.