The first factor of the class number of the p-th cyclotomic field

Kummer's conjecture states that the relative class number of the p-th cyclotomic field follows a strict asymptotic law. Granville has shown it unlikely to be true-it cannot be true if we assume the truth of two other widely believed conjectures. We establish a new bound for the error term in Kummer's conjecture, and more precisely we prove that , where beta is a possible Siegel zero of an odd