Trigonometric series and special values of L-functions

Inspired by representations of the class number of imaginary quadratic fields, in this paper, we give explicit evaluations of trigonometric series having generalized harmonic numbers as coefficients in terms of odd values of the Riemann zeta function and special values of L-functions subject to the parity obstruction. The coefficients that arise in these evaluations are shown to belong to certain cyclotomic extensions. Furthermore, using best polynomial approximation of smooth functions under uniform convergence due to Jackson and their log-sine integrals, we provide approximations of real numbers by combinations of special values of L-functions corresponding to the Legendre symbol. Our method for obtaining these results rests on a careful study of generating functions on the unit circle involving generalized harmonic numbers and the Legendre symbol, thereby relating them to values of polylogarithms and then finally extracting Fourier series of special functions that can be expressed in terms of Clausen functions.