Seeing the invisible: around generalized Kubert functions

Abstract. In this note, we shall make a survey on the generalized Kubert functions (also called functions with distribution property) emphasizing the use of the polylogarithm function in number theory. We shall reveal that the Bernoulli function part often appears in the form of a difference of the first polylogarithm function and elucidate the limiting behavior of functions in question as they approach a rational point on the unit circle. We shall clarify this elimination of Clausen function part in Riemann's posthumous fragment II and Mordell's Lambert-like series.