Special values of the Riemann zeta function capture all real numbers
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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.References
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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
- Received by editor(s): April 21, 2014
- Published electronically: February 26, 2015
- Communicated by: Ken Ono
- © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3359566