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
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Proc. Amer. Math. Soc. 143 (2015), 3743-3752 Request permission

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.

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