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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.

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