On the class numbers of cyclotomic fields

Let g_n denote the first factor of the class number of the nth cyclotomic field. It is proved that if n runs through a sequence of prime powers p^r tending to infinity, then

$\text{log}\text{g}{}_{\mathrm{n}}\text{～}\text{1}\text{4}\text{[1 ? (}\text{1}\text{p}\text{)]n}\text{log}\text{n}$

.