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On the size of an expression in the Nyman-Beurling-Báez-Duarte criterion for the Riemann Hypothesis

  • Helmut Maier,
    Department of Mathematics, University of Ulm, Helmholtzstrasse 18, 89081 Ulm, Germany
  • Michael Th. Rassias,
    Institute of Mathematics, University of Zurich, CH-8057, Zurich, Switzerland
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Abstract

A crucial role in the Nyman-Beurling-Báez-Duarte approach to the Riemann Hypothesis is played by the distance \[ d_N^2:=\inf_{A_N}\frac{1}{2\pi}\int_{-\infty}^\infty \left|1-\zeta A_N \left(\frac{1}{2}+it \right) \right|^2\frac{dt}{\frac{1}{4}+t^2}\:, \] where the infimum is over all Dirichlet polynomials $$A_N(s)=\sum_{n=1}^{N}\frac{a_n}{n^s}$$ of length $N$. In this paper we investigate $d_N^2$ under the assumption that the Riemann zeta function has four non-trivial zeros off the critical line.
Keywords: Riemann hypothesis, Riemann zeta function, Nyman-Beurling-Báez-Duarte criterion Riemann hypothesis, Riemann zeta function, Nyman-Beurling-Báez-Duarte criterion
MSC Classifications: 30C15, 11M26 show english descriptions Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10}
Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
30C15 - Zeros of polynomials, rational functions, and other analytic functions (e.g. zeros of functions with bounded Dirichlet integral) {For algebraic theory, see 12D10; for real methods, see 26C10}
11M26 - Nonreal zeros of $\zeta (s)$ and $L(s, \chi)$; Riemann and other hypotheses
 

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