Two Arguments that the Nontrivial Zeros of the Riemann Zeta Function are Irrational. Part II
Cardinal Stefan Wyszynski University
Faculty of Mathematics and Natural Sciences, College of Sciences
Wóycickiego 1/3, PL-01-938 Warsaw, Poland
E-mail: m.wolf@uksw.edu.pl
Received:
Received: 29 May 2020; revised: 18 June 2020; accepted: 18 June 2020; published online: 28 June 2020
DOI: 10.12921/cmst.2020.0000015
Abstract:
We extend the results of our previous computer experiment performed on the first 2600 nontrivial zeros γl of the Riemann zeta function calculated with 1000 digits accuracy to the set of 40 000 first zeros given with 40 000 decimal digits accuracy. We calculated the geometric means of the denominators of continued fractions expansions of these zeros and for all cases we get values very close to the Khinchin’s constant, which suggests that γl are irrational. Next we have calculated the n-th square roots of the denominators Qn of the convergents of the continued fractions obtaining values very close to the Khinchin-Lévy constant, again supporting the common opinion that γl are irrational.
Key words:
continued fractions, irrationality and normality of numbers, Khinchin and Levy constant, zeros of the Riemann’s zeta function
References:
[1] H.M. Edwards, Riemann’s zeta function, Pure and Applied Mathematics 58, Academic Press (1974).
[2] E.C. Titchmarsh, The Theory of the Riemann Zeta-function, The Clarendon Press Oxford University Press, New York, sec. ed. (1986).
[3] A. LeClair, An Electrostatic Depiction of the Validity of the Riemann Hypothesis and a Formula for the N-th Zero at Large N , International Journal of Modern Physics A 28, 1350151 (2013).
[4] F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark, NIST handbook of mathematical functions, NIST and CUP (2010).
[5] M. Wolf, Two arguments that the nontrivial zeros of the Rie- mann zeta function are irrational, Computational Methods in Science and Technology 24(4), 215–220 (2018).
[6] S.R. Finch, Mathematical Constants, Cambridge University Press (2003).
[7] A.Y. Khinchin, Continued Fractions, Dover Publications, New York (1997).
[8] C. Ryll-Nardzewski, On the ergodic theorems II (Ergodic theory of continued fractions), Studia Mathematica 12, 74–79 (1951).
[9] G. Beliakov, Y. Matiyasevich, Zeroes of Riemann’s zeta func- tion on the critical line with 40 000 decimal digits accuracy (2013).
[10] Zeroes of Riemann’s zeta function on the critical line with 40 000 decimal digits accuracy (2013). Available from: http://dro.deakin.edu.au/view/DU:30056270.
[11] G. Beliakov, Y. Matiyasevich, Approximation of Riemann’s zeta function by finite dirichlet series: A multiprecision nu- merical approach, Experimental Mathematics 24(2), 150–161 (2015).
[12] PARI/GP version 2.11.2 64 bits (2019). Available from http://pari.math.u-bordeaux.fr.
[13] P. Elliott, The Riemann Zeta function and coin tossing, Jour- nal für die reine und angewandte Mathematik 254, 100–109 (1972).
[14] K. Ford, A. Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function, Journal für die reine und angewandte Mathematik 579, 145–158 (2005).
[15] K. Ford, K. Soundararajan, A. Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function, II, Mathematische Annalen 343(3), 487–505 (2009).
We extend the results of our previous computer experiment performed on the first 2600 nontrivial zeros γl of the Riemann zeta function calculated with 1000 digits accuracy to the set of 40 000 first zeros given with 40 000 decimal digits accuracy. We calculated the geometric means of the denominators of continued fractions expansions of these zeros and for all cases we get values very close to the Khinchin’s constant, which suggests that γl are irrational. Next we have calculated the n-th square roots of the denominators Qn of the convergents of the continued fractions obtaining values very close to the Khinchin-Lévy constant, again supporting the common opinion that γl are irrational.
Key words:
continued fractions, irrationality and normality of numbers, Khinchin and Levy constant, zeros of the Riemann’s zeta function
References:
[1] H.M. Edwards, Riemann’s zeta function, Pure and Applied Mathematics 58, Academic Press (1974).
[2] E.C. Titchmarsh, The Theory of the Riemann Zeta-function, The Clarendon Press Oxford University Press, New York, sec. ed. (1986).
[3] A. LeClair, An Electrostatic Depiction of the Validity of the Riemann Hypothesis and a Formula for the N-th Zero at Large N , International Journal of Modern Physics A 28, 1350151 (2013).
[4] F.W.J. Olver, D.W. Lozier, R.F. Boisvert, C.W. Clark, NIST handbook of mathematical functions, NIST and CUP (2010).
[5] M. Wolf, Two arguments that the nontrivial zeros of the Rie- mann zeta function are irrational, Computational Methods in Science and Technology 24(4), 215–220 (2018).
[6] S.R. Finch, Mathematical Constants, Cambridge University Press (2003).
[7] A.Y. Khinchin, Continued Fractions, Dover Publications, New York (1997).
[8] C. Ryll-Nardzewski, On the ergodic theorems II (Ergodic theory of continued fractions), Studia Mathematica 12, 74–79 (1951).
[9] G. Beliakov, Y. Matiyasevich, Zeroes of Riemann’s zeta func- tion on the critical line with 40 000 decimal digits accuracy (2013).
[10] Zeroes of Riemann’s zeta function on the critical line with 40 000 decimal digits accuracy (2013). Available from: http://dro.deakin.edu.au/view/DU:30056270.
[11] G. Beliakov, Y. Matiyasevich, Approximation of Riemann’s zeta function by finite dirichlet series: A multiprecision nu- merical approach, Experimental Mathematics 24(2), 150–161 (2015).
[12] PARI/GP version 2.11.2 64 bits (2019). Available from http://pari.math.u-bordeaux.fr.
[13] P. Elliott, The Riemann Zeta function and coin tossing, Jour- nal für die reine und angewandte Mathematik 254, 100–109 (1972).
[14] K. Ford, A. Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function, Journal für die reine und angewandte Mathematik 579, 145–158 (2005).
[15] K. Ford, K. Soundararajan, A. Zaharescu, On the distribution of imaginary parts of zeros of the Riemann zeta function, II, Mathematische Annalen 343(3), 487–505 (2009).
