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

TL;DR

This paper refines the understanding of the class number of the $p$-th cyclotomic field by establishing a new bound on the error term in Kummer's conjecture, considering the influence of potential Siegel zeros.

Contribution

It provides a new bound for the error term in Kummer's conjecture, explicitly incorporating the effect of a possible Siegel zero, advancing the theoretical understanding of cyclotomic field class numbers.

Findings

Derived a precise asymptotic formula for $ ext{log}(h_p^-)$

Incorporated the impact of potential Siegel zeros into the error bounds

Enhanced the theoretical framework for analyzing cyclotomic field class numbers

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.