Inventiones Mathematicae

Invent math 86, 563-576 (1986) IflVQflttOfieS^

mathemattcae

© Springer-Verlag 1986

Simple zeros of the zeta function of a quadratic number field I

J . B . Conrey*\ A. Ghosh* \ and S.M. Gonek**^

^ Oklahoma State University, Department of Mathematics, Stillwater, OK 74078, USA ^ University of Rochester, Department of Mathematics, Rochester, NY 14620, USA

1 . Introduction

Let Ck(s) be the Dedekind zeta function of a quadratic extension К of the rationals. It is known that all the non-real zeros of Ск are in the strip 0 < Re. s<\ and that the number of zeros of ï^^ in the rectangle 0t={s = o^-iv. 0<ö-<l, 0<t<T} is asymptotically 7r"^TlogT. It is generally believed that all the zeros of Cx ^^"^ simple zeros, but it is not known whether Cx ^^^ infinitely many non-real simple zeros. In fact, prior to 1970 it was not even known that the Riemann zeta function C(s) has infinitely many complex simple zeros. The only related theorem was Selberg's result of 1942 that a positive portion of the zeros of ï^{s) are of odd order and lie on the critical line. In 1972, Montgomery showed that if the Riemann Hypothesis is true then at least 2/3 of the zeros of C(s) are simple and in 1974 Levinson proved that at least 1/3 of the zeros lie on the critical line. It was observed (independently) by Heath-Brown and Selberg that Levinson's work actually implied that 1/3 of the zeros are simple and on the critical line.

Thus , Levinson's method is the only method to date which gives ditional results on simple zeros of Dirichlet series. It is rather unlikely that his method could be successfully applied to Сх(Ю- Here we develop a new method and prove

Theorem . The number of simple zeros of Ск(^) ^^ ^ exceeds T^ if T is sufficiently large.

The zeros we find are also simple zeros of C{s) and we prove slightly more,

namely that the number of simple zeros exceeds T^~^ for any e>0 and

T>Tq ( 8 ) where ^ . ^ ^

_ J 1 (l + l6c4-16c2)^-l-4cl ,..

ö = max<^-——-,--------------------------------V (1

l - l - 6c 4c

* Work supported in part by NSF grant DMS-8401284 ** Work supported in part by NSF grant DMS-8503778