Kummer's incorrect conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field,
$h_1(p)$
, is further examined. Whereas Kummer conjectured that
$h_1(p) \sim G(p) := 2p(p/4\pi^2)^{(p-1)/4}$
it is shown, under certain plausible assumptions, that there exist constants
$a_\alpha, b_\alpha$
such that
$h_1(p) \sim \alpha G(p)$
for
$\sim a_\alpha x/\log^{b_\alpha} x$
primes
$p \le x$
whenever
$\log \alpha$
is rational. On the other hand, there are
$\ll_A x/\log^A x$
such primes when
$\log \alpha$
is irrational. Under a weak assumption it is shown that there are roughly the conjectured number of prime pairs
$p, mp\pm 1$
if and only if there are
$\gg_m x/\log^2 x$
primes
$p \le x$
for which
$h_1(p) \sim e^{\pm 1/2m} G(p)$
.