Approximation by special values of harmonic zeta function and log-sine integrals

Motivated by applications of log-sine integrals to a wide range of mathematical and physical problems, it is shown that real numbers and certain types of log-sine integrals can be strongly approximated by linear combinations of special values of the harmonic zeta function with the property that the coefficients belonging to these combinations turn out to be universal in the sense of being independent of special values. The approximation of real numbers by combinations of special values is reminiscent of the classical Diophantine approximation of Liouville numbers by rationals. Moreover, explicit representations of some specific log-sine integrals are obtained in terms of special values of the harmonic zeta function and the Riemann zeta function through a study of Fourier series involving harmonic numbers. In particular, special values of the harmonic zeta function and the less studied odd harmonic zeta function are expressed in terms of log-sine integrals over $[0, 2\pi]$ and $[0, \pi]$.

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Published 13 May 2014