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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.

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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

DOI: https://doi.org/10.1090/S0002-9939-2015-12649-4
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