On simple zeros of the Riemann zeta-function (Q2854224)

Let \(N(T)\) denote the number of zeros \(\rho = \beta + i\gamma\) of the Riemann zeta-function with \(0 < \gamma < T\), where each zero is counted with multiplicity. Let \(N^*(T)\) denote the number of such zeros that are simple and let \(N_d(T)\) denote the number of such distinct zeros. Define NEWLINE\[NEWLINE\kappa^* = \liminf_{T \to \infty} \frac{N^*(T)}{N(T)},\quad \kappa_d = \liminf_{T\to \infty} \frac{N_d(T)}{N(T)}.NEWLINE\]NEWLINE It is generally conjectured that all zeros of \(\zeta(s)\) are simple, that is \(\kappa ^* =1\). Unconditionally it is known that \(\kappa ^* \geq 0.4058\) and there are better bounds for \(\kappa ^*\) under different hypotheses. In particular \textit{J. B. Conrey, A. Ghosh} and \textit{S. M. Gonek} proved in [Proc. Lond. Math. Soc. (3) 76, No. 3, 497--522 (1998; Zbl 0907.11025)] that \(\kappa ^* \geq 19/27\) and \(\kappa_d \geq 0.84567\) by assuming both the Riemann Hypothesis and the generalized Lindelöf hypothesis. Readers are referred to the nicely written review of their paper in [Zbl 0907.11025]NEWLINENEWLINEIn the present paper, the authors obtained the same lower bound on \(\kappa ^*\) and the slight better bound \(\kappa _d \geq 0.84665\) by assuming only the Riemann Hypothesis. The main difficulty in the argument of the paper of Conrey et al. lies in estimating a sum of the form NEWLINE\[NEWLINE\sum_{q \sim Q}\,\, \mathop{{\sum}^*}_{\psi (\operatorname{mod}q)} \max_{M \leq X} \Big| \sum_{M \leq X} a(md) \psi(m) \Big|NEWLINE\]NEWLINE for natural number \(d\). It is natural to apply the maximal large sieve to this sum, by first applying Cauchy's inequality to turn the innermost sum into a square. However, the size of \(X\) is very large in comparison with \(Q\) and this necessitates the use of the generalized Lindelöf hypothesis in their argument. The main novel idea in the present paper is to use the second author's generalization of Vaughan's identity [Can. J. Math. 34, 1365--1377 (1982; Zbl 0478.10024)] to express the function \(a\) as a convolution and thereby replacing the innermost sum by a product of two Dirichlet polynomials of suitable lengths. This is made possible by the flexibility in Heath-Brown's identity. The last in applying Cauchy's inequality is then improved and hence the main results in this paper.