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 $h_1(p) \sim e^{\pm 1/2m} G(p)$ .

In this paper it is proved that, for x sufficiently large, every integer m with

can be written as m = Σ1≤n≤xεn/n, where εi, = 0 or 1.