Computation of the Kummer ratio of the class number for prime cyclotomic fields

Abstract

Let $\zeta_q$ be a primitive $q^{\text{th}}$ root of unity with $q$ an arbitrary odd prime. The ratio of Kummer's first factor of the class number of the cyclotomic number field $\mathbb{Q}(\zeta_q)$ and its expected order of magnitude (a simple function of $q$) is called the Kummer ratio and denoted by $r(q)$. It is known that typically $r(q)$ is close to 1, but nevertheless it is believed that it is unbounded, but only large on a very thin sequence of primes $q$. We propose an algorithm to compute $r(q)$ requiring the evaluation of $O(q\log q)$ products and $O(q)$ logarithms. Using it we obtain a new record maximum for $r(q)$, namely $r(6766811) =1.709379\dotsc$ (the old record being $r(5231)=1.556562\dotsc$). The program used and the results described here, are collected at the following address \url{http://www.math.unipd.it/~languasc/rq-comput.html}. This is a (preliminary) report about the computational part of a joint project with Pieter Moree, Sumaia Saad Eddin, and Alisa Sedunova.