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 $\beta$ is a possible Siegel zero of an $L(s,\chi )$, $\chi$ odd.

8 pages