On the class number of p-th cyclotomic field

Let h^-(p)h^-(p) be the relative class number of the p-th cyclotomic field. We show that logh^-(p) = (p+3)/4] logp - (p)/2] log2p+ log(1-b) + O(log₂² p)\log h^-(p) = {{p+3} \over {4}} \log p - {{p} \over {2}} \log 2\pi + \log (1-\beta ) + O(\log _2^2 p), where b\beta denotes a Siegel zero, if such a zero exists and p o -1 mod 4p\equiv -1\pmod {4}. Otherwise this term does not appear.