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.
Similar content being viewed by others
References
T. Estermann, Introduction to Modern Prime Number Theory, volume 41 of Cambridge Tracts in Mathematics and Mathematical Physics. Cambridge University Press, 1961.
H. Kadiri, Régions explicites sans zéros pour les fonctions l de dirichlet, phd thesis.
T. Lepistö, On the growth of the first factor of the class number of the prime cyclotomic field, Ann. Acad. Sci. Fenn. Ser. A I, (577):21, 1974.
Louboutin S.R.: Mean values of L-functions and relative class numbers of cyclotomic fields, Publ. Math. Debrecen, 78, 647–658 (2011)
J. M. Masley and H. L, Montgomery, Cyclotomic fields with unique factorization, J. Reine Angew. Math. 286/287 (1976), 248–256.
H. L. Montgomery, Topics in Multiplicative Number Theory, volume 227 of Lecture Notes in Mathematics, Springer-Verlag, 1971.
Montgomery H.L., Vaughan R.C.: The large sieve. Mathematika, 20, 119–134 (1973)
Puchta J.-C.: On the class number of p-th cyclotomic field. Arch. Math. (Basel) 74, 266–268 (2000)
L. C. Washington, Introduction to Cyclotomic Fields, volume 83 of Graduate Texts in Mathematics, Springer, 1982.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Debaene, K. The first factor of the class number of the p-th cyclotomic field. Arch. Math. 102, 237–244 (2014). https://doi.org/10.1007/s00013-014-0626-4
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-014-0626-4