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

Maier, Helmut ; Rassias, Michael Th.

Canadian mathematical bulletin, Tome 61 (2018) no. 3, pp. 622-627

Résumé

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.

Détail

DOI : 10.4153/CMB-2017-070-3

Mots-clés : 30C15, 11M26, Riemann hypothesis, Riemann zeta function, Nyman–Beurling–Bá-Duarte criterion