A simple limit formula for the relative class number h_p^-
by
Kurt Girstmair
Institut für Mathematik, Universität Innsbruck, Austria

Let p be a prime number ≥ 3 and k the largest natural number such that 2^k divides p-1. For an integer k put k^*=0 if p\DIV k; otherwise, let k^* be an arbitrary inverse of k mod p (so kk^* ≡ 1 mod p). Moreover, let m denote the Möbius function. Then

Ap= 2k+1p p ∞ å k=1  m(k) k sin 2p k* p

(*)

is a rational number closely connected with the relative class number h_p^- of the pth cyclotomic field \Q(e^2p i/p). For example, if h_p^- is > 1, squarefree, and prime to p-1 (there are 22 primes p < 200 fulfilling this condition), then

Ap= bp hp- ,

where b_p is a nonzero integer prime to h_p^-. Suppose we know, in addition, that h_p^- has ≤ d decimal digits. If we compute the right side of (*) with a precision of 2d+1 digits, we quickly obtain b_p and h_p^- from the continued fraction expansion of this approximate value of A_p.

Date received: March 17, 1999