L - Functions and Class Numbers of Imaginary Quadratic Fields and of Quadratic Extensions of an Imaginary Quadratic Field

Starting from the analytic class number formula involving its Lfunction, we first give an expression for the class number of an imaginary quadratic field which, in the case of large discriminants, provides us with a much more powerful numerical technique than that of counting the number of reduced definite positive binary quadratic forms, as has been used by Buell in order to compute his class number tables. Then, using class field theory, we will construct a periodic character x . defined on the ring of integers of a field K that is a quadratic extension of a principal imaginary quadratic field k , such that the zeta function of K is the product of the zeta function of k and of the L-function L(s, x) • We will then determine an integral representation of this L-function that enables us to calculate the class number of K numerically, as soon as its regulator is known. It will also provide us with an upper bound for these class numbers, showing that Hua's bound for the class numbers of imaginary and real quadratic fields is not the best that one could expect. We give statistical results concerning the class numbers of the first 50000 quadratic extensions of Q(() with prime relative discriminant (and with K/Q a non-Galois quartic extension). Our analytic calculation improves the algebraic calculation used by Lakein in the same way as the analytic calculation of the class numbers of real quadratic fields made by Williams and Broere improved the algebraic calculation consisting in counting the number of cycles of reduced ideals. Finally, we give upper bounds for class numbers of K that is a quadratic extension of an imaginary quadratic field k which is no longer assumed to be of class number one. 1. Class numbers of imaginary quadratic fields Let k be an imaginary quadratic field with discriminant D < 0 and character X ■ The analytic class number formula for this field is ¿71 Knowing the functional equations satisfied by the zeta function of k and the Riemann zeta function, one can easily deduce the functional equation satisfied by their quotient Lis, x). ie> F(s) = Pi 1 ~ s) with Fis) d¿f (ï^ï) SV(s)L(2s -l,x) = j"aWj > Received by the editor February 12, 1991 and, in revised form, May 17, 1991. 1991 Mathematics Subject Classification. Primary 11R29; Secondary 11R16. © 1992 American Mathematical Society 0025-5718/92 $1.00+ $.25 per page