Simple zeros of the zeta function of a quadratic number field. I
Using the work of \textit{N. Levinson} [Adv. Math. 13, 383-436 (1974; Zbl 0281.10017)], the reviewer showed [Bull. Lond. Math. Soc. 11, 17-18 (1979; Zbl 0409.10027)] that at least 1/3 of the zeros of the Riemann zeta- function \(\zeta\) (s) are simple and lie on the critical line. The authors present a new approach to the problem of simple zeros. With it they show that the Dedekind zeta-function \(\zeta_ K(s)\) of a quadratic number field has at least \(T^{6/11}\) simple zeros in the box B given by \(0<\sigma <1\), \(0<t\leq T\), for large T. In fact these zeros are all simple zeros of \(\zeta\) (s), too. In a forthcoming paper the authors hope to prove, on the Riemann Hypothesis, that a positive proportion of the zeros of \(\zeta_ K(s)\) are simple. To prove their result the authors factorize \(\zeta_ K(s)\) as \(\zeta (s)L(s,\psi),\) where \(\psi\) is a real Dirichlet character, and they observe that a zero \(\rho\) of \(\zeta\) (s) is a simple zero of \(\zeta_ K(s)\) providing that \(\zeta '(s)L(1-\rho,\psi)\neq 0.\) Cauchy's inequality then shows that the number of simple zeros in B is at least \(\Sigma^ 2_ 1 \Sigma_ 2^{-1}\), where \[ \Sigma_ 1=\sum_{\rho \in B}\zeta '(\rho)L(1-\rho,\psi),\quad \Sigma_ 2=\sum_{\rho \in B}| \zeta '(\rho)L(1-\rho,\psi) |^ 2, \] The proof is then completed by proving that \(\Sigma_ 1\gg T \log T\) and \(\Sigma_ 2\ll T^{16/11}\).
scientific article; zbMATH DE number 3975048
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