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

摘要

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 log(h_p~-) = (p+3)/4 log p + p/2 log(2π) + log(1 - β) + O(log_2 p), where β is a possible Siegel zero of an L(s, χ), χ odd.