Computation of the first factor of the class number of cyclotomic fields
In this interesting paper, the authors introduce a new method for computing the first factor \(h_ 1(p)\) of the class number of the \(p\)th cyclotomic field \((p\) an odd prime). As is well known, \(h_ 1(p)\) grows very fast, with about the same speed as the number \(G(p)=2p(p/4\pi^ 2)^{(p-1)/4}\). Following previous computations by \textit{D. H. Lehmer} and \textit{J. M. Masley} [Math. Comput. 32, 577--582 (1978; Zbl 0382.12003)], the authors first factorize \(h_ 1(p)\) into ``relative class numbers \(h_ e\) of degree \(e\)'', where \(e\) divides \(p-1\). When computing \(h_ e\) they use a trick which considerably shortens the procedure. They have computed \(h_ 1(p)\) in the range \(100<p<3000\) and, moreover, decomposed \(h_ 1(p)\) into factors which are often primes. The paper includes several tables extracted from the results (in full deposited in the UMT file) and there is a discussion concerning such topics as the divisibility by 2-powers of \(h_ 1(p)\) and the classical conjectures related to \(h_ 1(p)\). To the extensive bibliography given by the authors, one could add the papers by \textit{K. Iwasawa} [Nagoya Math. J. 27, 239--247 (1966; Zbl 0139.28104)] and the reviewer [Ann. Univ. Turku., Ser. A I 149, 1--8 (1971; Zbl 0215.07204)] both of which contain explicit information about the 2- divisibility of \(h_ 1(p)\). Note also the recent progress in the numerical checking of some of the conjectures mentioned; see the paper by \textit{J. P. Buhler}, \textit{R. E. Crandall}, \textit{R. Ernvall} and \textit{T. Metsänkylä} [Math. Comput. 61, No. 203, 151--153 (1993; Zbl 0789.11020)]. The authors also sketch arguments giving reasons to doubt the validity of Kummer's conjecture that \(h_ 1(p)\) be asymptotically equal to \(G(p)\). These arguments are developed further in a paper by the second author [Invent. Math. 100, No. 2, 321--338 (1990; Zbl 0701.11051)]. Moreover the authors show how an extension of recent work by D. M. Goldfeld, B. Gross and D. Zagier might be used to establish that \(h_ 1(p)\) is monotone increasing as \(p\geq 19\). \{A final remark by the reviewer: The analytic expression (4) of \(h_ 1(p)\) has of course been known long before the appearance of H. Hasse's monograph\}.
scientific article; zbMATH DE number 98393
scientific article