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Journal of the London Mathematical Society 2002 66(3):579-591; doi:10.1112/S0024610702003514
© 2002 by London Mathematical Society
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© The London Mathematical Society

Unit Fractions and the Class Number of a Cyclotomic Field

Ernest S. Croot, III and Andrew Granville

Department of Mathematics, University of California Berkeley, CA 94720-3840, USA, ecroot{at}math.berkeley.edu
Department of Mathematics, University of Georgia Athens, GA 30602, USA, andrew{at}math.uga.edu

Received 28 November 2000. Revision received 7 January 2002.

Kummer's incorrect conjectured asymptotic estimate for the size of the first factor of the class number of a cyclotomic field, h1(p), is further examined. Whereas Kummer conjectured that h1(p) ~ 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 h1(p) ~ {alpha}G(p) for ~ a{alpha}x/logb{alpha} x primes p ≤ x whenever log {alpha} is rational. On the other hand, there are <<A x/logA 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±1 if and only if there are >>m x/log2 x primes p ≤ x for which h1(p) ~ e±1/2mG(p).


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