Some remarks on the Báez-Duarte criterion for the Riemann Hypothesis
Cardinal Stefan Wyszynski University
Faculty of Mathematics and Natural Sciences. College of Sciences
ul. Wóycickiego 1/3, Auditorium Maximum, (room 113)
PL-01-938 Warsaw, Poland
e-mail: m.wolf@uksw.edu.pl
Received:
Received: 4 March 2014; revised: 9 April 2014; accepted: 11 April 2014; published online: 25 April 2014
DOI: 10.12921/cmst.2014.20.02.39-47
Abstract:
In this paper we are going to describe the results of the computer experiment, which in principle can rule out validity of the Riemann Hypothesis (RH). We use the sequence ck appearing in the Báez-Duarte criterion for the RH and compare two formulas for these numbers. We describe the mechanism of possible violation of the Riemann Hypothesis. Next we calculate c100000 with a thousand digits of accuracy using two different formulas for ck with the aim to disprove the Riemann Hypothesis in the case these two numbers will differ. We found the discrepancy only on the 996th decimal place (accuracy of 10−996 ). The computer experiment reported herein can be of interest for developers of Mathematica and PARI/GP.
Key words:
References:
[1] K. Appel and W. Haken, Every planar map is four colorable.
Part I. Discharging, Illinois J. Math., pages 429-490, (1977).
[2] K. Appel and W. Haken, Every planar map is four colorable.
Part II. Reducibility, Illinois J. Math., pages 491-567, (1977).
[3] T.C. Hales, The Kepler Conjecture, the series of pa-
pers: math.MG/9811071, math.MG/9811072, math.MG/
9811079, math.MG/9811079, for current status see:
http://sites.google.com/site/thalespitt/, http://www.math.pitt.
edu/ thales/kepler98/.
[4] P. Borwein, S. Choi, B. Rooney, and A. W eirathmueller,
The Riemann Hypothesis: A Resource For The Afficionado
And Virtuoso Alike, p. 137, Springer Verlag, 2007.
[5] A.M. Turing, Some calculations of the Riemann zeta-
function, Proc. London Math. Soc. (3) 3, 99-117, (1953).
[6] X. Gourdon, The 1013 First Zeros of the Riemann Zeta Func-
tion, and Zeros Computation at Very Large Height, Oct. 24,
2004, http://numbers.computation.free.fr/Constants/Misc-
ellaneous/zetazeros1e13-1e24.pdf.
[7] A.M. Odlyzko, The 1020 -th zero of the Riemann zeta func-
tion and 175 million of its neighbors, 1992 revision of 1989
manuscript.
[8] A.M. Odlyzko, The 1021-st zero of the Riemann zeta func-
tion, Nov. 1998 note for the informal proceedings of the
Sept. 1998 conference on the zeta function at the Edwin
Schroedinger Institute in Vienna.
[9] A.M. Odlyzko, The 1022-nd zero of the Riemann zeta func-
tion, In M. van Frankenhuysen and M. L. Lapidus, editors,
Dynamical, Spectral, and Arithmetic Zeta Functions, number
290 in Amer. Math. Soc., Contemporary Math. series, pages
139-144, 2001.
[10] J. Derbyshire, Prime Obsession. Bernhard Riemann and the
greatest unsolved problem in mathematics, p. 358, Joseph
Henry Press, Washington, 2003.
[11] A. Ivi` c, On some reasons for doubting the Riemann hypothe-
sis, arXiv:math/0311162, (November 2003).
[12] C.B. Haselgrove, A Disproof of a Conjecture of Polya, Math-
ematika 5, 141-145, (1958).
[13] R.S. Lehman, On Liouville’s Function, Math. Comput. 4,
311-320, (1960).
[14] A.M. Odlyzko and H.J.J. te Riele, Disproof of the Mertens
Conjecture, J. Reine Angew. Math. 357, 138-160, (1985).
[15] J. Pintz, An effective disproof of the Mertens conjecture, As-
terisque 147-148, 325-333, (1987).
[16] T. Kotnik and H. te Riele, The Mertens Conjecture Revisited,
In 7-th ANTS, volume 4076 of Lecture Notes in Computer
Science, pages 156-167, 2006.
[17] S.R. Finch, Mathematical Constants, Cambridge University
Press, 2003.
[18] A.M. Odlyzko, An improved bound for the de Bruijn–
Newman constant, Numerical Algorithms 25(1), 293-303,
(2000).
[19] D. Zagier, e-mail from 14 October 2009.
[20] X.J. Li, The Positivity of a Sequence of Numbers and the Rie-
mann Hypothesis, Journal of Number Theory 65(2), 325-333,
(1997).
[21] K. Ma ́ slanka, An Explicit Formula Relating Stieltjes Con-
stants and Li’s Numbers, 2004, http://xxx.lanl.gov/abs/
0406312.
[22] K. Ma ́slanka, Li’S Criterion For The Riemann Hypothesis
– Numerical Approach, Opuscula Mathematica 24, 103-114,
(2004).
[23] K. Ma ́slanka, Effective method of computing Li’s coefficients
and their properties, 2004, http://xxx.lanl.gov/abs/0402168.
[24] J.C. Lagarias, An elementary problem equivalent to the
Riemann Hypothesis, Amer. Math. Monthly 109, 534-543,
(2002).
[25] K. Briggs, Abundant Numbers and the Riemann Hypothesis,
Experimental Mathematics 15, Number 2, 251-256, (2006).
[26] G. Robin, Grandes valeurs de la fonction somme des di-
viseurs et Hypothèse de Riemann, J. Math. Pures Appl.
(9)63(2), 187-213, (1984).
[27] E.C. Titchmarsh, The Theory of the Riemann Zeta-function,
The Clarendon Press Oxford University Press, New York,
second edition, 1986, Edited and with a preface by D. R.
Heath-Brown.
[28] M. Wolf, Evidence in favor of the Baez-Duarte criterion for
the Riemann Hypothesis, Computational Methods in Science
and Technology 14, 47, (Nov 2008).
[29] K. Ma ́ slanka, A hypergeometric-like Representation of Zeta-
function of Riemann, Cracow Observatory preprint no.
1997/60, 1997, posted at arXiv: math-ph/0105007, (2001),
http://xxx.lanl.gov/abs/math/0105007.
[30] K. Ma ́ slanka, The Beauty of Nothingness: Essay on the Zeta
Function of Riemann, Acta Cosmologica XXIII-1, 13-17,
(1998).
[31] L. Baez-Duarte, On Maslanka’s representation for the Rie-
mann zeta-function, International Journal of Mathematics
and Mathematical Sciences 2010, 1-9, (2010).
[32] L. Báez-Duarte, A sequential Riesz-like criterion for the Rie-
mann Hypothesis, International Journal of Mathematics and
Mathematical Sciences 2005(21), 3527-3537, (2005).
[33] K. Ma ́slanka, Baez-Duarte’s Criterion for the Riemann Hy-
pothesis and Rice’s Integrals, math.NT/0603713, (2006).
[34] M. Abramowitz and I.A. Stegun, Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical Tables,
Dover, New York, ninth Dover printing, tenth GPO printing
edition, 1964.
[35] N. Ng, Extreme values of ζ (ρ), arXiv:math/0706.1765v1
[math.NT], (2007).
[36] The PARI Group, Bordeaux, PARI/GP, version 2.3.2, 2008,
available from http://pari.math.u-bordeaux.fr/.
[37] A.M. Odlyzko, Tables of zeros of the Riemann zeta function,
http://www.dtc.umn.edu/ odlyzko/zeta_tables/index.html.
[38] T.M. Apostol, Introduction to Analytic Number Theory,
Springer-Verlag, New York, 1976, Undergraduate Texts in
Mathematics.
[39] H. Cohen, F.R. Villegas, and D. Zagier, Convergence accel-
eration of alternating series, Experiment. Math. 9(1), 3-12,
(2000).
[40] J.M. Borwein and P.B. Borwein, Strange Series and High
Precision Fraud, The American Mathematical Monthly
99(7), 622-640, (1992).
[41] C.S. Unnikrishnan and G.T. Gillies, The electrical neutral-
ity of atoms and of bulk matter, Metrologia 41(5), 125-135,
(2004).
In this paper we are going to describe the results of the computer experiment, which in principle can rule out validity of the Riemann Hypothesis (RH). We use the sequence ck appearing in the Báez-Duarte criterion for the RH and compare two formulas for these numbers. We describe the mechanism of possible violation of the Riemann Hypothesis. Next we calculate c100000 with a thousand digits of accuracy using two different formulas for ck with the aim to disprove the Riemann Hypothesis in the case these two numbers will differ. We found the discrepancy only on the 996th decimal place (accuracy of 10−996 ). The computer experiment reported herein can be of interest for developers of Mathematica and PARI/GP.
Key words:
References:
[1] K. Appel and W. Haken, Every planar map is four colorable.
Part I. Discharging, Illinois J. Math., pages 429-490, (1977).
[2] K. Appel and W. Haken, Every planar map is four colorable.
Part II. Reducibility, Illinois J. Math., pages 491-567, (1977).
[3] T.C. Hales, The Kepler Conjecture, the series of pa-
pers: math.MG/9811071, math.MG/9811072, math.MG/
9811079, math.MG/9811079, for current status see:
http://sites.google.com/site/thalespitt/, http://www.math.pitt.
edu/ thales/kepler98/.
[4] P. Borwein, S. Choi, B. Rooney, and A. W eirathmueller,
The Riemann Hypothesis: A Resource For The Afficionado
And Virtuoso Alike, p. 137, Springer Verlag, 2007.
[5] A.M. Turing, Some calculations of the Riemann zeta-
function, Proc. London Math. Soc. (3) 3, 99-117, (1953).
[6] X. Gourdon, The 1013 First Zeros of the Riemann Zeta Func-
tion, and Zeros Computation at Very Large Height, Oct. 24,
2004, http://numbers.computation.free.fr/Constants/Misc-
ellaneous/zetazeros1e13-1e24.pdf.
[7] A.M. Odlyzko, The 1020 -th zero of the Riemann zeta func-
tion and 175 million of its neighbors, 1992 revision of 1989
manuscript.
[8] A.M. Odlyzko, The 1021-st zero of the Riemann zeta func-
tion, Nov. 1998 note for the informal proceedings of the
Sept. 1998 conference on the zeta function at the Edwin
Schroedinger Institute in Vienna.
[9] A.M. Odlyzko, The 1022-nd zero of the Riemann zeta func-
tion, In M. van Frankenhuysen and M. L. Lapidus, editors,
Dynamical, Spectral, and Arithmetic Zeta Functions, number
290 in Amer. Math. Soc., Contemporary Math. series, pages
139-144, 2001.
[10] J. Derbyshire, Prime Obsession. Bernhard Riemann and the
greatest unsolved problem in mathematics, p. 358, Joseph
Henry Press, Washington, 2003.
[11] A. Ivi` c, On some reasons for doubting the Riemann hypothe-
sis, arXiv:math/0311162, (November 2003).
[12] C.B. Haselgrove, A Disproof of a Conjecture of Polya, Math-
ematika 5, 141-145, (1958).
[13] R.S. Lehman, On Liouville’s Function, Math. Comput. 4,
311-320, (1960).
[14] A.M. Odlyzko and H.J.J. te Riele, Disproof of the Mertens
Conjecture, J. Reine Angew. Math. 357, 138-160, (1985).
[15] J. Pintz, An effective disproof of the Mertens conjecture, As-
terisque 147-148, 325-333, (1987).
[16] T. Kotnik and H. te Riele, The Mertens Conjecture Revisited,
In 7-th ANTS, volume 4076 of Lecture Notes in Computer
Science, pages 156-167, 2006.
[17] S.R. Finch, Mathematical Constants, Cambridge University
Press, 2003.
[18] A.M. Odlyzko, An improved bound for the de Bruijn–
Newman constant, Numerical Algorithms 25(1), 293-303,
(2000).
[19] D. Zagier, e-mail from 14 October 2009.
[20] X.J. Li, The Positivity of a Sequence of Numbers and the Rie-
mann Hypothesis, Journal of Number Theory 65(2), 325-333,
(1997).
[21] K. Ma ́ slanka, An Explicit Formula Relating Stieltjes Con-
stants and Li’s Numbers, 2004, http://xxx.lanl.gov/abs/
0406312.
[22] K. Ma ́slanka, Li’S Criterion For The Riemann Hypothesis
– Numerical Approach, Opuscula Mathematica 24, 103-114,
(2004).
[23] K. Ma ́slanka, Effective method of computing Li’s coefficients
and their properties, 2004, http://xxx.lanl.gov/abs/0402168.
[24] J.C. Lagarias, An elementary problem equivalent to the
Riemann Hypothesis, Amer. Math. Monthly 109, 534-543,
(2002).
[25] K. Briggs, Abundant Numbers and the Riemann Hypothesis,
Experimental Mathematics 15, Number 2, 251-256, (2006).
[26] G. Robin, Grandes valeurs de la fonction somme des di-
viseurs et Hypothèse de Riemann, J. Math. Pures Appl.
(9)63(2), 187-213, (1984).
[27] E.C. Titchmarsh, The Theory of the Riemann Zeta-function,
The Clarendon Press Oxford University Press, New York,
second edition, 1986, Edited and with a preface by D. R.
Heath-Brown.
[28] M. Wolf, Evidence in favor of the Baez-Duarte criterion for
the Riemann Hypothesis, Computational Methods in Science
and Technology 14, 47, (Nov 2008).
[29] K. Ma ́ slanka, A hypergeometric-like Representation of Zeta-
function of Riemann, Cracow Observatory preprint no.
1997/60, 1997, posted at arXiv: math-ph/0105007, (2001),
http://xxx.lanl.gov/abs/math/0105007.
[30] K. Ma ́ slanka, The Beauty of Nothingness: Essay on the Zeta
Function of Riemann, Acta Cosmologica XXIII-1, 13-17,
(1998).
[31] L. Baez-Duarte, On Maslanka’s representation for the Rie-
mann zeta-function, International Journal of Mathematics
and Mathematical Sciences 2010, 1-9, (2010).
[32] L. Báez-Duarte, A sequential Riesz-like criterion for the Rie-
mann Hypothesis, International Journal of Mathematics and
Mathematical Sciences 2005(21), 3527-3537, (2005).
[33] K. Ma ́slanka, Baez-Duarte’s Criterion for the Riemann Hy-
pothesis and Rice’s Integrals, math.NT/0603713, (2006).
[34] M. Abramowitz and I.A. Stegun, Handbook of Mathematical
Functions with Formulas, Graphs, and Mathematical Tables,
Dover, New York, ninth Dover printing, tenth GPO printing
edition, 1964.
[35] N. Ng, Extreme values of ζ (ρ), arXiv:math/0706.1765v1
[math.NT], (2007).
[36] The PARI Group, Bordeaux, PARI/GP, version 2.3.2, 2008,
available from http://pari.math.u-bordeaux.fr/.
[37] A.M. Odlyzko, Tables of zeros of the Riemann zeta function,
http://www.dtc.umn.edu/ odlyzko/zeta_tables/index.html.
[38] T.M. Apostol, Introduction to Analytic Number Theory,
Springer-Verlag, New York, 1976, Undergraduate Texts in
Mathematics.
[39] H. Cohen, F.R. Villegas, and D. Zagier, Convergence accel-
eration of alternating series, Experiment. Math. 9(1), 3-12,
(2000).
[40] J.M. Borwein and P.B. Borwein, Strange Series and High
Precision Fraud, The American Mathematical Monthly
99(7), 622-640, (1992).
[41] C.S. Unnikrishnan and G.T. Gillies, The electrical neutral-
ity of atoms and of bulk matter, Metrologia 41(5), 125-135,
(2004).
