An upper bound of class number of cyclotomic field \(Q(\zeta_ p)\) (Q802665)

The author proves that the class number of the pth cyclotomic field (p an odd prime) is less than \(10(\pi p/16)^{(p-2)/2}.\) Previously \textit{I. Sh. Slavutskij} [Zap. Nauchn. Semin. Leningr. Otd. Mat. Inst. Steklova 154, 136-143 (1986; Zbl 0607.12004)] obtained a slightly weaker result. Both authors' arguments follow similar lines, but the present paper contains a different technical trick in the treatment of the explicit expression of L(1,\(\chi\)), the value of Dirichlet's L-function for an even character \(\chi\). Another novelty is the use of \textit{K. Feng}'s upper bound for the relative class number [Proc. Am. Math. Soc. 84, No.4, 479-482 (1982; Zbl 0493.12009)]. This bound may indeed suit best to the present purpose but the reviewer would like to remark that, contrary to the author's claim, it is not the best result known. A better estimate has been proved by \textit{J. M. Masley} and \textit{H. L. Montgomery} [J. Reine Angew. Math. 286/287, 248-256 (1976; Zbl 0335.12013)].