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A lower bound in an approximation problem involving the zeros of the Riemann zeta function. (English) Zbl 1029.11045
This paper is a contribution to the Buerling-Nyman formulation of the Riemann hypothesis as an approximation problem. Let $$K =L^2( (0,\infty), dt)$$, let $$\chi$$ be the characteristic function of the interval $$(0,1]$$, and let $$\rho$$ be the fractional part function. For $$0 < \lambda <1$$, let $$\mathcal{B}_\lambda$$ be the subspace of $$K$$ consisting of finite linear combinations of the functions $$t\mapsto \rho(\theta/t)$$ for $$\lambda \leq \theta \leq 1$$. The Nyman-Beurling theorem is that the Riemann Hypothesis is true if and only if $$\chi$$ is in the closure of the union of the $$\mathcal{B}_\lambda$$, $$0 < \lambda <1$$.
Let $$D(\lambda)$$ denote the Hilbert-space distance $$\inf_{f\in \mathcal{B}_\lambda} \|\chi-f\|$$. L. Báez-Duarte, M. Balazard, B. Landreau, and E. Saias [Adv. Math. 149, 130-144 (2000; Zbl 1008.11032)] proved that $$\liminf_{\lambda \to 0} D(\lambda) (\log(1/\lambda))^{1/2} \geq (\sum_{\rho} 1/|\rho|^2)^{1/2},$$ where the sum on the right-hand side is over non-trivial zeros $$\rho$$ of the Riemann zeta-function with each zero being counted only once, regardless of multiplicity. In this paper, the author shows that the right-hand side of the above may be replaced by $$(\sum_{\rho} m_\rho^2/|\rho|^2)^{1/2},$$ where $$m_\rho$$ is the multiplicity of the zero $$\rho$$. The proof depends on a construction of certain Hilbert space vectors that are used to control $$D(\lambda)$$.

##### MSC:
 11M26 Nonreal zeros of $$\zeta (s)$$ and $$L(s, \chi)$$; Riemann and other hypotheses
##### Keywords:
Riemann zeta-function
Full Text:
##### References:
  Báez-Duarte, L., A class of invariant unitary operators, Adv. in math., 144, 1-12, (1999) · Zbl 0978.47025  Báez-Duarte, L.; Balazard, M.; Landreau, B.; Saias, E., Notes sur la fonction ζ de Riemann 3, Adv. in math., 149, 130-144, (2000) · Zbl 1008.11032  Balazard, M.; Saias, E., The nyman-Beurling equivalent form for the Riemann hypothesis, Exposition math., 18, 131-138, (2000) · Zbl 0954.11029  Beurling, A., A closure problem related to the Riemann zeta-function, Proc. nat. acad. sci., 41, 312-314, (1955) · Zbl 0065.30303  Burnol, J.-F., Sur LES formules explicites I: analyse invariante, C. R. acad. sci. Paris Sér., I 331, 423-428, (2000) · Zbl 0992.11064  Burnol, J.-F., Scattering on the p-adic field and a trace formula, Internat. math. res. notes, 2000, 57-70, (2000) · Zbl 1038.11033  Burnol, J.-F., An adelic causality problem related to abelian L-functions, J. number theory, 87, 253-269, (2001) · Zbl 1038.11057  Burnol, J.-F., Quaternionic gamma functions and their logarithmic derivatives as spectral functions, Math. res. lett., 8, 209-223, (2001) · Zbl 1024.11075  Burnol, J.-F., Sur certains espaces de Hilbert de fonctions entières, liés à la transformation de Fourier et aux fonctions L de Dirichlet et de Riemann, C. R. acad. sci. Paris Sér., I 333, 201-206, (2001) · Zbl 1057.11039  A. Cauchy, Oeuvres complètes d’Augustin Cauchy, Publiées sous la direction scientifique de l’Académie des Sciences, IIe Série, Gauthier-Villars, Paris. · JFM 19.0019.01  Connes, A., Formule de trace en géométrie non-commutative et hypothèse de Riemann, C. R. acad. sci. Paris Sér., I 323, 1231-1236, (1996) · Zbl 0864.46042  Connes, A., Trace formula in non-commutative geometry and the zeros of the Riemann zeta function, Selecta math. (N.S.), 5, 29-106, (1999) · Zbl 0945.11015  Ehm, W., A family of probability densities related to the Riemann zeta function, () · Zbl 1045.11059  Grenander, U.; Rosenblatt, M., An extension of a theorem of G. szegö and its application to the study of stochastic processes, Trans. amer. math. soc., 76, 112-126, (1954) · Zbl 0059.11804  Hoffman, K., Banach spaces of analytic functions, (1962), Prentice-Hall, Inc Englewood Cliffs · Zbl 0117.34001  Nevai, P., Géza freud, orthogonal polynomials and Christoffel functions. A case study, J. approx. theory, 48, 3-167, (1986) · Zbl 0606.42020  B. Nyman, On the One-Dimensional Translation Group and Semi-Group in Certain Function Spaces, Thesis, University of Uppsala, 55p, 1950. · Zbl 0037.35401  Tate, J., Fourier analysis in number fields and Hecke’s zeta function, Princeton, 1950, ()
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