Special values of the Riemann zeta function capture all real numbers

Alkan, Emre

doi:10.1090/S0002-9939-2015-12649-4

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
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Abstract: It is shown that the set of odd values

$\displaystyle \{\zeta (3), \zeta (5),\dots , \zeta (2k+1),\dots \}$

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.

[1] Emre Alkan, On Dirichlet 𝐿-functions with periodic coefficients and Eisenstein series, Monatsh. Math. 163 (2011), no. 3, 249–280. MR 2805873, https://doi.org/10.1007/s00605-010-0211-2
[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, https://doi.org/10.1016/j.jnt.2011.02.013
[3] Emre Alkan, Values of Dirichlet 𝐿-functions, Gauss sums and trigonometric sums, Ramanujan J. 26 (2011), no. 3, 375–398. MR 2860694, https://doi.org/10.1007/s11139-010-9292-8
[4] Emre Alkan, On linear combinations of special values of 𝐿-functions, Manuscripta Math. 139 (2012), no. 3-4, 473–494. MR 2974286, https://doi.org/10.1007/s00229-011-0526-x
[5] Emre Alkan, Averages of values of 𝐿-series, Proc. Amer. Math. Soc. 141 (2013), no. 4, 1161–1175. MR 3008864, https://doi.org/10.1090/S0002-9939-2012-11506-0
[6] Emre Alkan, Series representations in the spirit of Ramanujan, J. Math. Anal. Appl. 410 (2014), no. 1, 11–26. MR 3109816, https://doi.org/10.1016/j.jmaa.2013.08.021
[7] 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, https://doi.org/10.4310/CNTP.2013.v7.n3.a5
[8] E. Alkan, Series representing transcendental numbers that are not -numbers, Int. J. Number Theory, to appear. doi: 10.1142/S1793042115500487
[9] 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, https://doi.org/10.1215/S0012-7094-01-10831-4
[10] Necdet Batir, Integral representations of some series involving 2𝑘\choose𝑘⁻¹𝑘⁻ⁿ and some related series, Appl. Math. Comput. 147 (2004), no. 3, 645–667. MR 2011078, https://doi.org/10.1016/S0096-3003(02)00802-0
[11] Necdet Batir, On the series ∑^{∞}_{𝑘=1}3𝑘\choose𝑘⁻¹𝑘⁻ⁿ𝑥^{𝑘}, Proc. Indian Acad. Sci. Math. Sci. 115 (2005), no. 4, 371–381. MR 2184197, https://doi.org/10.1007/BF02829799
[12] Bruce C. Berndt, Modular transformations and generalizations of several formulae of Ramanujan, Rocky Mountain J. Math. 7 (1977), no. 1, 147–189. MR 0429703, https://doi.org/10.1216/RMJ-1977-7-1-147
[13] 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, https://doi.org/10.1515/crll.1978.303-304.332
[14] Bruce C. Berndt, Ramanujan’s notebooks. Part I, Springer-Verlag, New York, 1985. With a foreword by S. Chandrasekhar. MR 781125
[15] Bruce C. Berndt, Ramanujan’s notebooks. Part II, Springer-Verlag, New York, 1989. MR 970033
[16] 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.
[17] H. Bohr, R. Courant, Neue Anwendungen der Theorie der Diophantischen Approximationen auf die Riemannsche Zetafunktion, J. Reine Angew. Math. 144 (1914), 249-274.
[18] 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, https://doi.org/10.1090/S0002-9939-02-06649-2
[19] Kenneth R. Davidson and Allan P. Donsig, Real analysis and applications, Undergraduate Texts in Mathematics, Springer, New York, 2010. Theory in practice. MR 2568574
[20] D. J. Griffiths, Introduction to Quantum Mechanics, Second Edition, Pearson, Prentice Hall, 2005.
[21] 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
[22] 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, https://doi.org/10.1023/A:1011449413387
[23] 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
[24] K. Mahler, Zur Approximation der Exponentialfunktionen und des Logarithmus, I, II, J. Reine Angew. Math. 166 (1932), 118-150.
[25] István Mező and Ayhan Dil, Hyperharmonic series involving Hurwitz zeta function, J. Number Theory 130 (2010), no. 2, 360–369. MR 2564902, https://doi.org/10.1016/j.jnt.2009.08.005
[26] S. Ramanujan, On the integral $\int _{0}^{x}\frac {\tan ^{-1}t}{t}\ dt$ , J. Indian Math. Soc. 7 (1915), 93-96.
[27] Linas Vepštas, On Plouffe’s Ramanujan identities, Ramanujan J. 27 (2012), no. 3, 387–408. MR 2901266, https://doi.org/10.1007/s11139-011-9335-9
[28] 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, https://doi.org/10.1090/S0002-9939-08-09622-6