One of the central themes in number theory is the study of special values of $L$-functions, in particular, the investigation of their transcendental nature and algebraic relations among them. A special function governing linear relations among the values $L(1,\chi)$ as $\chi$ varies over Dirichlet characters modulo $q$, is the digamma function, which is the logarithmic derivative of the gamma function. In this talk, we discuss the arithmetic nature and related properties of values of the digamma function at rational arguments, and emphasize their connection with a seemingly unrelated conjecture of Erdos, which is still open.