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On the Size of an Expression in the Nyman–Beurling-Báez–Duarte Criterion for the Riemann Hypothesis

Published online by Cambridge University Press:  20 November 2018

Helmut Maier
Department of Mathematics, University of Ulm, Helmholtzstrasse 18, 89081 Ulm, Germany, e-mail:
Michael Th. Rassias
Institute of Mathematics, University of Zurich, CH-8057, Zurich, Switzerland, e-mail :
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A crucial role in the Nyman-Beurling-Báez-Duarte approach to the Riemann Hypothesis is played by the distance

$$d_{N}^{2}:=\underset{{{A}_{N}}}{\mathop{\inf }}\,\frac{1}{2\pi }\int _{-\infty }^{\infty }{{\left| 1-\zeta {{A}_{N}}\left( \frac{1}{2}+it \right) \right|}^{2}}\frac{dt}{\frac{1}{4}+{{t}^{2}}},$$

where the infimum is over all Dirichlet polynomials

$${{A}_{N}}\left( s \right)\,=\,\sum\limits_{n=1}^{N}{\frac{{{a}_{n}}}{{{n}^{s}}}}$$

of length $N$. In this paper we investigate $d_{N}^{2}$ under the assumption that the Riemann zeta function has four nontrivial zeros off the critical line.

Research Article
Copyright © Canadian Mathematical Society 2018


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