Unit Fractions and the Class Number of a Cyclotomic Field

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 ) ∼ G ( p ):= 2 p ( p /4π 2 ) ( p −1)/4 it is shown, under certain plausible assumptions, that there exist constants a α , b α such that h 1 ( p ) ∼ α G ( p ) for ∼ a α x /log b α x primes p ⩽ x whenever log α is rational. On the other hand, there are ≪ A x /log A x such primes when log α is irrational. Under a weak assumption it is shown 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 ±1/2 m G ( p ).