Special values of the Riemann zeta function capture all real numbers
Author:
Emre Alkan
Journal:
Proc. Amer. Math. Soc. 143 (2015), 3743-3752
MSC (2010):
Primary 11M06, 41A50, 42A16
DOI:
https://doi.org/10.1090/S0002-9939-2015-12649-4
Published electronically:
February 26, 2015
MathSciNet review:
3359566
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: It is shown that the set of odd values \[ \{\zeta (3), \zeta (5),\dots , \zeta (2k+1),\dots \} \] of the Riemann zeta function is rich enough to capture real numbers in an approximation aspect. Precisely, we prove that any real number can be strongly approximated by certain linear combinations of these odd values, where the coefficients belonging to these combinations are universal in the sense of being independent of $\zeta (n)$ for all integers $n \geq 2$. This approximation property is reminiscent of the classical Diophantine approximation of Liouville numbers by rationals.
- Emre Alkan, On Dirichlet $L$-functions with periodic coefficients and Eisenstein series, Monatsh. Math. 163 (2011), no. 3, 249–280. MR 2805873, DOI https://doi.org/10.1007/s00605-010-0211-2
- Emre Alkan, On the mean square average of special values of $L$-functions, J. Number Theory 131 (2011), no. 8, 1470–1485. MR 2793888, DOI https://doi.org/10.1016/j.jnt.2011.02.013
- Emre Alkan, Values of Dirichlet $L$-functions, Gauss sums and trigonometric sums, Ramanujan J. 26 (2011), no. 3, 375–398. MR 2860694, DOI https://doi.org/10.1007/s11139-010-9292-8
- Emre Alkan, On linear combinations of special values of $L$-functions, Manuscripta Math. 139 (2012), no. 3-4, 473–494. MR 2974286, DOI https://doi.org/10.1007/s00229-011-0526-x
- Emre Alkan, Averages of values of $L$-series, Proc. Amer. Math. Soc. 141 (2013), no. 4, 1161–1175. MR 3008864, DOI https://doi.org/10.1090/S0002-9939-2012-11506-0
- Emre Alkan, Series representations in the spirit of Ramanujan, J. Math. Anal. Appl. 410 (2014), no. 1, 11–26. MR 3109816, DOI https://doi.org/10.1016/j.jmaa.2013.08.021
- Emre Alkan, Approximation by special values of harmonic zeta function and log-sine integrals, Commun. Number Theory Phys. 7 (2013), no. 3, 515–550. MR 3210726, DOI https://doi.org/10.4310/CNTP.2013.v7.n3.a5
- E. Alkan, Series representing transcendental numbers that are not $U$-numbers, Int. J. Number Theory, to appear. doi: 10.1142/S1793042115500487
- George E. Andrews, Jorge Jiménez-Urroz, and Ken Ono, $q$-series identities and values of certain $L$-functions, Duke Math. J. 108 (2001), no. 3, 395–419. MR 1838657, DOI https://doi.org/10.1215/S0012-7094-01-10831-4
- Necdet Batir, Integral representations of some series involving ${2k\choose k}^{-1}k^{-n}$ and some related series, Appl. Math. Comput. 147 (2004), no. 3, 645–667. MR 2011078, DOI https://doi.org/10.1016/S0096-3003%2802%2900802-0
- Necdet Batir, On the series $\sum ^\infty _{k=1}{3k\choose k}^{-1}k^{-n}x^k$, Proc. Indian Acad. Sci. Math. Sci. 115 (2005), no. 4, 371–381. MR 2184197, DOI https://doi.org/10.1007/BF02829799
- Bruce C. Berndt, Modular transformations and generalizations of several formulae of Ramanujan, Rocky Mountain J. Math. 7 (1977), no. 1, 147–189. MR 429703, DOI https://doi.org/10.1216/RMJ-1977-7-1-147
- Bruce C. Berndt, Analytic Eisenstein series, theta-functions, and series relations in the spirit of Ramanujan, J. Reine Angew. Math. 303(304) (1978), 332–365. MR 514690, DOI https://doi.org/10.1515/crll.1978.303-304.332
- Bruce C. Berndt, Ramanujan’s notebooks. Part I, Springer-Verlag, New York, 1985. With a foreword by S. Chandrasekhar. MR 781125
- Bruce C. Berndt, Ramanujan’s notebooks. Part II, Springer-Verlag, New York, 1989. MR 970033
- H. Bohr, Über Diophantische Approximationen und ihre Anwendungen auf Dirichlet’sche Reihen, besonders auf die Riemann’sche Zetafunktion, 5. Skand. Mat. Kongr. Helsingfors (1922), 131–154.
- H. Bohr, R. Courant, Neue Anwendungen der Theorie der Diophantischen Approximationen auf die Riemannsche Zetafunktion, J. Reine Angew. Math. 144 (1914), 249–274.
- Gwynneth H. Coogan and Ken Ono, A $q$-series identity and the arithmetic of Hurwitz zeta functions, Proc. Amer. Math. Soc. 131 (2003), no. 3, 719–724. MR 1937408, DOI https://doi.org/10.1090/S0002-9939-02-06649-2
- Kenneth R. Davidson and Allan P. Donsig, Real analysis and applications, Undergraduate Texts in Mathematics, Springer, New York, 2010. Theory in practice. MR 2568574
- D. J. Griffiths, Introduction to Quantum Mechanics, Second Edition, Pearson, Prentice Hall, 2005.
- Dunham Jackson, The theory of approximation, American Mathematical Society Colloquium Publications, vol. 11, American Mathematical Society, Providence, RI, 1994. Reprint of the 1930 original. MR 1451140
- Shigeru Kanemitsu, Hiroshi Kumagai, and Masami Yoshimoto, On rapidly convergent series expressions for zeta- and $L$-values, and log sine integrals, Ramanujan J. 5 (2001), no. 1, 91–104. MR 1829811, DOI https://doi.org/10.1023/A%3A1011449413387
- A. A. Karatsuba and S. M. Voronin, The Riemann zeta-function, De Gruyter Expositions in Mathematics, vol. 5, Walter de Gruyter & Co., Berlin, 1992. Translated from the Russian by Neal Koblitz. MR 1183467
- K. Mahler, Zur Approximation der Exponentialfunktionen und des Logarithmus, I, II, J. Reine Angew. Math. 166 (1932), 118–150.
- István Mező and Ayhan Dil, Hyperharmonic series involving Hurwitz zeta function, J. Number Theory 130 (2010), no. 2, 360–369. MR 2564902, DOI https://doi.org/10.1016/j.jnt.2009.08.005
- S. Ramanujan, On the integral $\int _{0}^{x}\frac {\tan ^{-1}t}{t}\ dt$, J. Indian Math. Soc. 7 (1915), 93–96.
- Linas Vepštas, On Plouffe’s Ramanujan identities, Ramanujan J. 27 (2012), no. 3, 387–408. MR 2901266, DOI https://doi.org/10.1007/s11139-011-9335-9
- Alexandru Zaharescu and Mohammad Zaki, An algebraic independence result for Euler products of finite degree, Proc. Amer. Math. Soc. 137 (2009), no. 4, 1275–1283. MR 2465649, DOI https://doi.org/10.1090/S0002-9939-08-09622-6
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Additional Information
Emre Alkan
Affiliation:
Department of Mathematics, Koç University, Rumelifeneri Yolu, 34450, Sarıyer, Istanbul, Turkey
Email:
ealkan@ku.edu.tr
Keywords:
Riemann zeta function,
special values,
odd values,
even values,
approximation property
Received by editor(s):
April 21, 2014
Published electronically:
February 26, 2015
Communicated by:
Ken Ono
Article copyright:
© Copyright 2015
American Mathematical Society