Around the Gauss theorem on the values of Euler's digamma function at rational points

We find representations of generating functions for the values of the Riemann zeta function at odd points and related numbers in terms of definite integrals of trigonometric functions depending on the parameter $a$. In particular, new integral representations for the Euler digamma function $\psi(a)$ are obtained. The resulting integrals are such that they can be calculated in terms of the hypergeometric series ${}_3F_{2}$ and ${}_4F_{3}$ for some values of the parameters and $z=1$. Moreover, if $a$ is a proper rational fraction, then they reduce to integrals of $R(\sin x, \cos x)$, where $R$ is a rational fractional function of two variables, and are explicitly calculated. In this case, various analogues of the Gauss theorem on the values of the function $\psi(a)$ at rational points and, in particular, one more proof of it are obtained.