Real quadratic fields with large class number - Springer Link

Math. Ann. 225, 173--176 (1977) © by Springer-Verlag 1977

Real Quadratic Fields with Large Class Number Hugh L. Montgomery and Peter J. Weinberger Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48104, USA

Let h, d, a, and R be the class number, discriminant, fundamental unit, and regulator, respectively, of the real quadratic field 11)(]/3). Let Z be the primitive quadratic character (mod d), and let

L(s, x)= ~. z(n) n-s. n=l

Then R = tog~, and

L(t, z ) = h R d -~ .

(1)

Since L(1, g)~logd, R>(½+o(1))logd, it follows that h~]//d. Moreover, Littlewood [4] showed that if all nontrivial zeros of L(s, Z) lie on the critical line Re s = 3, then L(1, Z) < (2e7 + o(1)) log logd. Hence h < (4e ~+ o(1))d ~(logd)- a log logd

(2)

assuming the Generalized Riemann Hypothesis. In this paper we show that the hypothetical estimate (2) can not be improved upon, apart from the value of the constant. Theorem. There is an absolute constant c > 0 such that

h > cd ~(logd)- l log logd

(3)

for infinitely many real quadratic felds ~ ( ~ l ) . To prove the Theorem we construct d for which R < l o g d , and L(1, Z)> c loglogd. Then (3) follows from (1). We consider only square-free d with d~_ 1(mod4),d = n 2 + 1.Then e = n + ~/d < 2 ] / ~ < d , and so R x~ such square-free dc 1 logy>cloglogd for almost all of these d. This completes the proof of the Theorem. References 1. Bombieri, E.: Le grand crible dans la th6orie analytique des nombres. Ast6risque 18, Socidt6 Math. de France 1974 2. Elliott, P. D. T, A. : The distribution of the quadratic class number. Litovsk. Mat. Sb. 10, 189--196 (1970) 3. Estermann,T.: Einige S~itze fiber quadratfreie Zahlen. Math. Ann. 105, 653-662 (193t) 4. Littlewood,J.E.: On the class number of the corpus P ( I ~ k ) . Proc. London Math. Soc. 27, 358--372 (1927) 5. Montgomery, H, L. : Zeros of L-functions. Invent. Math. 8, 346--354 (1969) 6. Montgomery, H. L. : Topics in multiplicative number theory, Lecture Notes in Mathematics 227. Berlin, Heidelberg, New York: Springer 1971 7. Titchmarsh, E. C. : The theory of the Riemann zeta-function. Oxford: Oxford University Press 195t Received July 6, 1976