Evidence in Favor of the Baez-Duarte Criterion for the Riemann Hypothesis
Institute of Theoretical Physics, University of Wroc aw
Pl. Maxa Borna 9, PL-50-204 Wroc aw, Poland
e-mail: mwolf@ift.uni.wroc.pl
Received:
Received; 22 October 2008; published online: 30 October 2008
DOI: 10.12921/cmst.2008.14.01.47-54
OAI: oai:lib.psnc.pl:645
Abstract:
We present the results of the numerical experiments in favor of the Baez-Duarte criterion for the Riemann Hypothesis. We give formulae allowing calculation of numerical values of the numbers ck appearing in this criterion for arbitrary large k. We present plots of ck for k c (1,109).
Key words:
References:
[1] K. Maslanka, A hypergeometric-like Representation of Zeta-function of Riemann, Cracow Observatory preprint no. 1997/60, 1997; K. Maslanka, A hypergeometric-like Representation of Zeta-function of Riemann, posted at arXiv:math-ph/0105007 2001.
[2] L. Baez-Duarte, On Maslankas representation for the Riemann zeta function, 2003, math.NT/0307214.
[3 L. Baez-Duarte, A new necessary and su±cient condition for the Riemann Hypothesis, 2003, math.NT/0307215.
[4] L. Baez-Duarte, A sequential Riesz-like criterion for the Riemann Hypothesis, International Journal of Mathematics and Mathematical Sciences 3527-3537 (2005).
[5] K. Ma lanka, Liczba i kwant (in English: Number and quant), OBI 2004 Kraków.
[6] K. Ma lanka, Baez-Duartes Criterion for the Riemann Hypothesis and Rices Integrals, math.NT/0603713 v2 1 Apr 2006.
[7] R. L. Graham, D. E. Knuth and O. Patashnik, Binomial Coefficients. Ch. 5 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.
[8] E. C. Titchmarsh, The Theory of the Riemann Zeta Function, 2nd ed. New York: Clarendon Press, 1987.
[9] M. Riesz, Sur l’hypothe`se de Riemann, Acta Math. 40, 185-190 (1916).
[10] J. Cislo and M. Wolf, Criteria equivalent to the Riemann Hypothesis, arXiv:math.NT/0808.0640v2.
[11] L. Baez-Duarte, Möbius-Convolutions and the Riemann Hypothesis, arXiv:math.NT/0504402
[12] PARI/GP, version 2.2.11, Bordeaux, 2005, http://pari.math.u-bordeaux.fr/.
[13] S. Beltraminelli and D. Merlini, The criteria of Riesz, Hardy-Littlewood et al. for the Riemann Hypothesis revisited using similar functions, math.NT/0601138.
[14] P. Flajolet and R. Sedgewick, Theoretical Computer Science, 144 (1-2), 101-124 (1995).
[15] E. C. Titchmarsh, The Theory of Functions, 2nd ed. Oxford, England: Oxford University Press, 1960.
[16] S. Beltraminelli and D. Merlini, Riemann Hypothesis: The Riesz-Hardy-Littlewood wave in the long wavelength region, math.NT/0605565
[17] A. M. Odlyzko, An improved bound for the de Bruijn-Newman constant, Numerical Algorithms 25, 293-303 (2000).
[18] X.-J. Li, The Positivity of a Sequence of Numbers and the Riemann Hypothesis, J. Number Th. 65, 325-333 (1997).
[19] J. Oesterle unpublished, cited in P. Biane, J. Pitman and M. Yor, Probability Laws Related to the Jacobi Theta and Riemann Zeta Functions, and Brownian Excursions. Bull. Amer. Math. Soc. 38, 435-465 (2001).
We present the results of the numerical experiments in favor of the Baez-Duarte criterion for the Riemann Hypothesis. We give formulae allowing calculation of numerical values of the numbers ck appearing in this criterion for arbitrary large k. We present plots of ck for k c (1,109).
Key words:
References:
[1] K. Maslanka, A hypergeometric-like Representation of Zeta-function of Riemann, Cracow Observatory preprint no. 1997/60, 1997; K. Maslanka, A hypergeometric-like Representation of Zeta-function of Riemann, posted at arXiv:math-ph/0105007 2001.
[2] L. Baez-Duarte, On Maslankas representation for the Riemann zeta function, 2003, math.NT/0307214.
[3 L. Baez-Duarte, A new necessary and su±cient condition for the Riemann Hypothesis, 2003, math.NT/0307215.
[4] L. Baez-Duarte, A sequential Riesz-like criterion for the Riemann Hypothesis, International Journal of Mathematics and Mathematical Sciences 3527-3537 (2005).
[5] K. Ma lanka, Liczba i kwant (in English: Number and quant), OBI 2004 Kraków.
[6] K. Ma lanka, Baez-Duartes Criterion for the Riemann Hypothesis and Rices Integrals, math.NT/0603713 v2 1 Apr 2006.
[7] R. L. Graham, D. E. Knuth and O. Patashnik, Binomial Coefficients. Ch. 5 in Concrete Mathematics: A Foundation for Computer Science, 2nd ed. Reading, MA: Addison-Wesley, 1994.
[8] E. C. Titchmarsh, The Theory of the Riemann Zeta Function, 2nd ed. New York: Clarendon Press, 1987.
[9] M. Riesz, Sur l’hypothe`se de Riemann, Acta Math. 40, 185-190 (1916).
[10] J. Cislo and M. Wolf, Criteria equivalent to the Riemann Hypothesis, arXiv:math.NT/0808.0640v2.
[11] L. Baez-Duarte, Möbius-Convolutions and the Riemann Hypothesis, arXiv:math.NT/0504402
[12] PARI/GP, version 2.2.11, Bordeaux, 2005, http://pari.math.u-bordeaux.fr/.
[13] S. Beltraminelli and D. Merlini, The criteria of Riesz, Hardy-Littlewood et al. for the Riemann Hypothesis revisited using similar functions, math.NT/0601138.
[14] P. Flajolet and R. Sedgewick, Theoretical Computer Science, 144 (1-2), 101-124 (1995).
[15] E. C. Titchmarsh, The Theory of Functions, 2nd ed. Oxford, England: Oxford University Press, 1960.
[16] S. Beltraminelli and D. Merlini, Riemann Hypothesis: The Riesz-Hardy-Littlewood wave in the long wavelength region, math.NT/0605565
[17] A. M. Odlyzko, An improved bound for the de Bruijn-Newman constant, Numerical Algorithms 25, 293-303 (2000).
[18] X.-J. Li, The Positivity of a Sequence of Numbers and the Riemann Hypothesis, J. Number Th. 65, 325-333 (1997).
[19] J. Oesterle unpublished, cited in P. Biane, J. Pitman and M. Yor, Probability Laws Related to the Jacobi Theta and Riemann Zeta Functions, and Brownian Excursions. Bull. Amer. Math. Soc. 38, 435-465 (2001).
