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

Abstract

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 \({\log(h_p^-)=\frac{p+3}{4} \log p +\frac{p}{2}\log(2\pi)+\log(1-\beta)+O(\log_2 p)}\), where β is a possible Siegel zero of an \({L(s,\chi), \chi}\) odd.