We show how the classical polylogarithm function \(\textrm{Li}_s(z)\) and its relatives, the Hurwitz zeta function and the Lerch function are all of a spectral nature, and can explain many properties of the complex powers of the Laplacian on the circle and of the distribution \((x+i 0)^s\) . We also make a relation with a result of Keiper [Fractional Calculus and its relationship to Riemann’s zeta function, Master of Science, Ohio State University, Mathematics (1975)].
