A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis

Luis Báez-Duarte

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (2003)

  • Volume: 14, Issue: 1, page 5-11
  • ISSN: 1120-6330

Abstract

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According to the well-known Nyman-Beurling criterion the Riemann hypothesis is equivalent to the possibility of approximating the characteristic function of the interval 0 , 1 in mean square norm by linear combinations of the dilations of the fractional parts 1 / a ⁢ x for real a greater than 1 . It was conjectured and established here that the statement remains true if the dilations are restricted to those where the a ’s are positive integers. A constructive sequence of such approximations is given.

How to cite

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Báez-Duarte, Luis. "A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 14.1 (2003): 5-11. <http://eudml.org/doc/252348>.

@article{Báez2003,
abstract = {According to the well-known Nyman-Beurling criterion the Riemann hypothesis is equivalent to the possibility of approximating the characteristic function of the interval $(0,1]$ in mean square norm by linear combinations of the dilations of the fractional parts $\\{1/ax\\}$ for real $a$ greater than $1$. It was conjectured and established here that the statement remains true if the dilations are restricted to those where the $a$’s are positive integers. A constructive sequence of such approximations is given.},
author = {Báez-Duarte, Luis},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Riemann zeta function; Riemann hypothesis; Nyman-Beurling theorem},
language = {eng},
month = {3},
number = {1},
pages = {5-11},
publisher = {Accademia Nazionale dei Lincei},
title = {A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis},
url = {http://eudml.org/doc/252348},
volume = {14},
year = {2003},
}

TY - JOUR
AU - Báez-Duarte, Luis
TI - A strengthening of the Nyman-Beurling criterion for the Riemann hypothesis
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 2003/3//
PB - Accademia Nazionale dei Lincei
VL - 14
IS - 1
SP - 5
EP - 11
AB - According to the well-known Nyman-Beurling criterion the Riemann hypothesis is equivalent to the possibility of approximating the characteristic function of the interval $(0,1]$ in mean square norm by linear combinations of the dilations of the fractional parts $\{1/ax\}$ for real $a$ greater than $1$. It was conjectured and established here that the statement remains true if the dilations are restricted to those where the $a$’s are positive integers. A constructive sequence of such approximations is given.
LA - eng
KW - Riemann zeta function; Riemann hypothesis; Nyman-Beurling theorem
UR - http://eudml.org/doc/252348
ER -

References

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  3. BÁEZ-DUARTE, L., Arithmetical versions of the Nyman-Beurling criterion for the Riemann hypothesis. IJMMS, IJMMS/1324, 2002, to appear. Zbl1069.11037MR1926809DOI10.1155/S0161171202013248
  4. BÁEZ-DUARTE, L. - BALAZARD, M. - LANDREAU, B. - SAIAS, E., Notes sur la fonction ζ de Riemann, 3. Adv. in Math., 149, n. 1, 2000, 130-144. Zbl1008.11032MR1742356DOI10.1006/aima.1999.1861
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  7. BURNOL, J.F., A lower bound in an approximation problem involving the zeroes of the Riemann zeta function. Math., AIM01/048, 2001Adv. in Math., to appear. Zbl1029.11045MR1929303DOI10.1006/aima.2001.2066
  8. BURNOL, J.F., On an analytic estimate in the theory of the Riemann Zeta function and a theorem of Báez-Duarte. Preprint, 18 Feb. 2002, available at arXiv.org as math.NT/0202166. MR2136671
  9. CONREY, J.B. - MYERSON, G., On the Balazard-Saias criterion for the Riemann hypothesis. Posted in http://arXiv.org/abs/math.NT/002254, Feb. 2000. 
  10. VAN FRANKENHUYSEN, M., Zero-Free Regions for the Riemann Zeta-Function, density of invariant Subspaces of Functions, and the Theory of equal Distribution. Preprint, 1997. Zbl1034.11045
  11. LANDREAU, B. - RICHARD, F., Le critère de Beurling et Nyman pour l’hypothèse de Riemann: aspects numériques. Université de Bordeaux, preprint 2001, submitted to Experimental Mathematics. Zbl1117.11305MR1959747
  12. LEE, J., Convergence and the Riemann Hypothesis. Comm. Korean Math. Soc., 11, 1996, 57-62. Zbl0941.11029MR1430941
  13. NIKOLSKI, N., Distance formulae and invariant subspaces, with an application to localization of zeroes of the Riemann ζ -function. Ann. Inst. Fourier (Grenoble), 45, 1995, n. 1, 1-17. Zbl0816.30026MR1324128
  14. NYMAN, B., On some groups and semigroups of translations. Thesis, Uppsala1950. 
  15. RADEMACHER, H., Topics in Analytic Number Theory. Die Grundleheren der mathematischen Wissenschaften, Band 169, Springer-Verlag, New York1973. Zbl0253.10002MR364103
  16. TITCHMARSH, E.C., The Theory of the Riemann Zeta-Function. Clarendon Press, Oxford1951. Zbl0042.07901MR46485
  17. VASYUNIN, V.I., Sur un système biorthogonal relié a l’hypothèse de Riemann (in Russian). Algebra i Annaliz, 7, 1995, 118-135. Also appeared as: On a biorthogonal system related with the Riemann hypothesis, St. Petersburg Math. J., 7, 1996, 405-419. Zbl0851.11051MR1334152
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