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

  • Helmut Maier (a1) and Michael Th. Rassias (a2)

Abstract

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.

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[1] Bäez-Duarte, L., Balazard, M., Landreau, B., and Saias, E., Notes sur lafonction f de Riemann. III. Adv. Math. 149(2000), no. 1, 130144. http://dx.doi.Org/10.1006/aima.1999.1861
[2] Bäez-Duarte, L., Balazard, M., Landreau, B., and Saias, E., Etüde de Vautocorrelation multiplicative de lafonction ‘partiefractionnaire'. Ramanujan J. 9(2005), no. 1-2, 215240. http://dx.doi.Org/10.1007/s11139-005-0834-4
[3] Bettin, S., Conrey, J. B., and Farmer, D. W., An optimal choice of Dirichlet polynomiah for the Nyman-Beurling criterion. Proc. Steklov Inst. Math. 280(2013), suppl. 2, S30-S36. http://dx.doi.Org/10.1134/S0081543813030036
[4] Burnol, J. F., A lower bound in an approximation problem involving the zeros of the Riemann zeta function. Adv. Math. 170(2002), 56-70. http://dx.doi.Org/10.1006/aima.2001.2066
[5] Titchmarsh, E. C., The theory of the Riemann Zeta-function. Second ed., The Clarendon Press, Oxford University Press, New York, 1986.
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