Lengths of the periods of the continued fraction expansion of quadratic irrationalities and on the class numbers of real quadratic fields

Abstract

The fundamental result of the paper is the following theorem: suppose that the Riemann conjecture is valid for the Dedekind ~-functions of all fields G( __i_ _(I+s ~ ~/k)-Then there exists a constant C > 0 such that on the interval p ~ x one can find at least Cx log -I x prime numbers p for which h(Sp 2) = 2. Here h(d) is the number of proper equivalence classes of primitive binary quadratic forms of discriminant d. In addition, it is proved that ~ ~(Sp 2) s p = 0 Cx ~/2) p4x For these sequence of discriminants of a special form with increasing square-free part, one has obtained a nontrivial estimate from above for the number of classes.The Gauss--Hasse conjecture on the existence of an infinite number of real quadratic fields of the same class is a striking example of a problem in which number theory has made very little progress since Gauss' time. Lately a number of works have been published in which this conjecture is formulated in a more precise (containing quantitative statements) form (see [i], [2]), but, just as before, something definite can be said about the number h(d) of classes of binary quadratic forms of discriminant d (d > 0, d is not a perfect square) only in those cases when the fundamental unit e(d) of the corresponding order is explicitly computed. An exception is the paper [3], where the considered problem is formulated in terms which go beyond the framework of the present investigation. The unique tool for the computation of ~(d) is the expansion of ~d in a continued fraction. It is known [4], [5] that, roughly speaking, log e(d) coincides with the length s of the period of this expansion (more precisely, the difference between them is negligibly small in comparison with the abyss which separates the existing results from the real state of affairs; see Sec. 3). The quantity s behaves itself in an extremely irregular manner (see Table 1 in Sec. 3). In this sense it reminds us of the length of the period of the expansion of a rational number into a decimal fraction. We note that this analogy is not terminological since both problems are connected with the problemof primitive roots. In the case of decimal fractions the answer to the problem (belonging to Gauss) is given by Artin's conjecture, while in the case of a continued fraction for discriminants of a special form (for example, d = 5p =, p is prime) by its analogue for the corresponding quadratic field (see Sec. 2, where Hooley's results are carried over to the case of the field ~(~) ).

A positive density of fundamental discriminants with large regulator

“…74, 85;Fouvry and Jouve 2012, Theorem 2]. See also [Golubeva 1987] for the study of the case d = 5 p 2 . However none of these works manages to produce an infinite family of squarefree d's.…”