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

We introduce an algorithm to compute $r(q)$, the ratio between $h_1(q)$, Kummer's emph{first factor of the class number} of $Q(zeta_q)$, and its expected order of magnitude $G(q)$, where $q$ is an odd prime number and $zeta_q$ is a primitive $q$-root of unity. Such an algorithm requires $Odi{qlog q}$ products together with $Odi{q}$ logarithm evaluations. We obtain a new maximum for $r(q)$, namely $r(6766811) =1.709379042dotsc$, see section ef{chi-Bernoulli-method} below. The program used and the results here described are collected at the following address \url{http://www.math.unipd.it/~languasc/rq-comput.html}.