Unit Fractions and the Class Number of a Cyclotomic Field

We further examine Kummer's incorrect conjectured asymptotic estimate for the size of the rst factor of the class number of a cyclotomic eld, h 1 (p). Whereas Kummer had conjectured that h 1 (p) G(p) := 2p(p=4 2) p?1 4 we show, under certain plausible assumptions, that there exist constants a ; b such that h 1 (p) G(p) for a x= log b x primes p x whenever log is rational. On the other hand, there are A x= log A x such primes when log is irrational. Under a weak assumption we show that there are roughly the conjectured number of prime pairs p; mp 1 if and only if there are m x= log 2 x primes p x for which h 1 (p) e 1=2m G(p).