Pseudo-differential operators on the circle, Bernoulli polynomials

We show how the classical polylogarithm function Liₛ(z) 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 ( 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)].