and Multiplication formulas for the non-principal characters may be given in the form of Dirichlet L-functions. Periodic zeta function The periodic zeta function is sometimes defined as where Lis(z) is the polylogarithm. It obeys the duplication formula As such, it is an eigenvector of the Bernoulli operator with eigenvalue 2−s. The multiplication theorem is The periodic zeta function occurs in the reflection formula for the Hurwitz zeta function, which is why the relation that it obeys, and the Hurwitz zeta relation, differ by the interchange of s → −s. The Bernoulli polynomials may be obtained as a limiting case of the periodic zeta function, taking s to be an integer, and thus the multiplication theorem there can be derived from the above. Similarly, substituting q = log z leads to the multiplication theorem for the polylogarithm. Polylogarithm The duplication formula takes the form The general multiplication formula is in the form of a Gauss sum or discrete Fourier transform: These identities follow from that on the periodic zeta function, taking z = log q. Kummer's function The duplication formula for Kummer's function is and thus resembles that for the polylogarithm, but twisted by i. Bernoulli polynomials For the Bernoulli polynomials, the multiplication theorems were given by Joseph Ludwig Raabe in 1851: and for the Euler polynomials, and The Bernoulli polynomials may be obtained as a special case of the Hurwitz zeta function, and thus the identities follow from there. Bernoulli map The Bernoulli map is a certain simple model of a dissipative dynamical system, describing the effect of a shift operator on an infinite string of coin-flips (the Cantor set). The Bernoulli map is a one-sided version of the closely related Baker's map. The Bernoulli map generalizes to a k-adic version, which acts on infinite strings of k symbols: this is the Bernoulli scheme. The transfer operator corresponding to the shift operator on the Bernoulli scheme is given by Perhaps not surprisingly, the eigenvectors of this operator are given by the Bernoulli polynomials. That is, one has that It is the fact that the eigenvalues that marks this as a dissipative system: for a non-dissipative measure-preserving dynamical system, the eigenvalues of the transfer operator lie on the unit circle. One may construct a function obeying the multiplication theorem from any totally multiplicative function. Let be totally multiplicative; that is, for any integers m, n. Define its Fourier series as Assuming