Two Arguments that the Nontrivial Zeros of the Riemann Zeta Function are Irrational
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: 31 October 2018; revised: 19 November 2018; accepted: 21 November 2018; published online: 07 December 2018
DOI: 10.12921/cmst.2018.0000049
Abstract:
We have used the first 2600 nontrivial zeros γl of the Riemann zeta function calculated with 1000 digits accuracy and developed them into the continued fractions. We calculated the geometrical means of the denominators of these con- tinued fractions and for all cases we get values 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 close to the Khinchin-Lévy constant, again supporting the common opinion that γl are irrational.
Key words:
References:
[1] B. Conrey, H.M. Bui, M.P. Young, More than 41% of the zeros of the zeta function are on the critical line, Acta Arithmetica 150, 35–64 (2011).
[2] J.B.Conrey, More than two fifths of the zeros of the Riemann zeta function are on the critical line, J.Reine. Angew. Math. 399, 1–26 (1989).
[3] D. Cvijovic ́, J. Klinowski, Continued-Fraction Expansions for the Riemann Zeta Function and Polylogarithms, Proceedings of the American Mathematical Society 125(9), 2543–2550 (1997).
[4] P. Elliott, The Riemann Zeta function and coin tossing, Journal für die reine und angewandte Mathematik, 254, 100–109 (1972).
[5] S.R. Finch, Mathematical Constants, Cambridge University Press, 2003.
[6] 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).
[7] K. Ford and A. Zaharescu, On the distribution of imaginary parts of zeros of the riemann zeta function, Journal für die reine und angewandte Mathematik 2005, 03 (2005).
[8] X. Gourdon, The 1013 first zeros of the Riemann Zeta Function, and zeros computation at very large height, Oct. 24, 2004, http://numbers.computation.free.fr/Constants/ Miscellaneous/zetazeros1e13-1e24.pdf.
[9] G. H. Hardy, Sur les zéros de la fonction ζ(s) de Riemann, C. R. Acad. Sci. Paris 158, 1012–1014 (1914).
[10] A.E. Ingham, On two conjectures in the theory of numbers, Amer. J. Math. 64, 313–319 (1942).
[11] A.Y. Khinchin, Continued Fractions, Dover Publications, New York, 1997.
[12] N. Levinson, More than one third of zeros of Riemann’s zeta-function are on σ = 1/2, Advances in Math. 13, 383–436 (1974).
[13] A.M. Odlyzko, The 1020-th zero of the Riemann zeta function and 175 million of its neighbors, 1992 revision of 1989 manuscript.
[14] A.M. Odlyzko, The 1021-st zero of the Riemann zeta function, Nov. 1998 note for the informal proceedings of the Sept. 1998 conference on the zeta function at the Edwin Schroedinger Institute in Vienna.
[15] A.M. Odlyzko, Tables of zeros of the Riemann zeta function, http://www.dtc.umn.edu/ odlyzko/zeta_tables/index.html.
[16] A.M. Odlyzko, Primes, quantum chaos, and computers, Number Theory, National Research Council, pages 35–46, 1990.
[17] A.M. Odlyzko, The 1022-nd zero of the Riemann zeta function, In M. van Frankenhuysen, M.L. Lapidus, editors, Dynamical, Spectral, and Arithmetic Zeta Functions, number 290 in Amer. Math. Soc., Contemporary Math. series, pages 139–144, 2001.
[18] A.M. Odlyzko, H.J.J. te Riele, Disproof of the Mertens Conjecture, J. Reine Angew. Math. 357, 138–160 (1985).
[19] B. Riemann, Ueber die anzahl der primzahlen unter einer gegebenen grösse, Monatsberichte der Berliner Akademie, pages 671–680, November 1859, english translation available at http://www.maths.tcd.ie/pub/Hist Math/People/Riemann.
[20] C. Ryll-Nardzewski, On the ergodic theorems II (Ergodic theory of continued fractions), Studia Mathematica 12, 74–79 (1951).
[21] A. Selberg, On the zeros of Riemann’s zeta-function, Skr. Norske Vid. Akad. Oslo I. 10 59, (1942).
[22] C.L. Siegel, Transcendental Numbers, Princeton University Press, 1949.
[23] The PARI Group, Bordeaux, PARI/GP, version 2.3.2, 2008, available from http://pari.math.u-bordeaux.fr/.
[24] E.W. Weisstein, CRC Concise Encyclopedia of Mathematics, Chapman & Hall/CRC, 2009.
We have used the first 2600 nontrivial zeros γl of the Riemann zeta function calculated with 1000 digits accuracy and developed them into the continued fractions. We calculated the geometrical means of the denominators of these con- tinued fractions and for all cases we get values 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 close to the Khinchin-Lévy constant, again supporting the common opinion that γl are irrational.
Key words:
References:
[1] B. Conrey, H.M. Bui, M.P. Young, More than 41% of the zeros of the zeta function are on the critical line, Acta Arithmetica 150, 35–64 (2011).
[2] J.B.Conrey, More than two fifths of the zeros of the Riemann zeta function are on the critical line, J.Reine. Angew. Math. 399, 1–26 (1989).
[3] D. Cvijovic ́, J. Klinowski, Continued-Fraction Expansions for the Riemann Zeta Function and Polylogarithms, Proceedings of the American Mathematical Society 125(9), 2543–2550 (1997).
[4] P. Elliott, The Riemann Zeta function and coin tossing, Journal für die reine und angewandte Mathematik, 254, 100–109 (1972).
[5] S.R. Finch, Mathematical Constants, Cambridge University Press, 2003.
[6] 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).
[7] K. Ford and A. Zaharescu, On the distribution of imaginary parts of zeros of the riemann zeta function, Journal für die reine und angewandte Mathematik 2005, 03 (2005).
[8] X. Gourdon, The 1013 first zeros of the Riemann Zeta Function, and zeros computation at very large height, Oct. 24, 2004, http://numbers.computation.free.fr/Constants/ Miscellaneous/zetazeros1e13-1e24.pdf.
[9] G. H. Hardy, Sur les zéros de la fonction ζ(s) de Riemann, C. R. Acad. Sci. Paris 158, 1012–1014 (1914).
[10] A.E. Ingham, On two conjectures in the theory of numbers, Amer. J. Math. 64, 313–319 (1942).
[11] A.Y. Khinchin, Continued Fractions, Dover Publications, New York, 1997.
[12] N. Levinson, More than one third of zeros of Riemann’s zeta-function are on σ = 1/2, Advances in Math. 13, 383–436 (1974).
[13] A.M. Odlyzko, The 1020-th zero of the Riemann zeta function and 175 million of its neighbors, 1992 revision of 1989 manuscript.
[14] A.M. Odlyzko, The 1021-st zero of the Riemann zeta function, Nov. 1998 note for the informal proceedings of the Sept. 1998 conference on the zeta function at the Edwin Schroedinger Institute in Vienna.
[15] A.M. Odlyzko, Tables of zeros of the Riemann zeta function, http://www.dtc.umn.edu/ odlyzko/zeta_tables/index.html.
[16] A.M. Odlyzko, Primes, quantum chaos, and computers, Number Theory, National Research Council, pages 35–46, 1990.
[17] A.M. Odlyzko, The 1022-nd zero of the Riemann zeta function, In M. van Frankenhuysen, M.L. Lapidus, editors, Dynamical, Spectral, and Arithmetic Zeta Functions, number 290 in Amer. Math. Soc., Contemporary Math. series, pages 139–144, 2001.
[18] A.M. Odlyzko, H.J.J. te Riele, Disproof of the Mertens Conjecture, J. Reine Angew. Math. 357, 138–160 (1985).
[19] B. Riemann, Ueber die anzahl der primzahlen unter einer gegebenen grösse, Monatsberichte der Berliner Akademie, pages 671–680, November 1859, english translation available at http://www.maths.tcd.ie/pub/Hist Math/People/Riemann.
[20] C. Ryll-Nardzewski, On the ergodic theorems II (Ergodic theory of continued fractions), Studia Mathematica 12, 74–79 (1951).
[21] A. Selberg, On the zeros of Riemann’s zeta-function, Skr. Norske Vid. Akad. Oslo I. 10 59, (1942).
[22] C.L. Siegel, Transcendental Numbers, Princeton University Press, 1949.
[23] The PARI Group, Bordeaux, PARI/GP, version 2.3.2, 2008, available from http://pari.math.u-bordeaux.fr/.
[24] E.W. Weisstein, CRC Concise Encyclopedia of Mathematics, Chapman & Hall/CRC, 2009.
