Multiplication Theorem - Periodic Zeta Function

The periodic zeta function is sometimes defined as

F(s;q) = \sum_{m=1}^\infty \frac {e^{2\pi imq}}{m^s} =\operatorname{Li}_s\left(e^{2\pi i q} \right)

where Lis(z) is the polylogarithm. It obeys the duplication formula

2^{-s} F(s;q) = F\left(s,\frac{q}{2}\right) + F\left(s,\frac{q+1}{2}\right).

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.

Read more about this topic:  Multiplication Theorem

Famous quotes containing the words periodic and/or function:

    “But parents can be understanding and accept the more difficult stages as necessary times of growth for the child. Parents can appreciate the fact that these phases are not easy for the child to live through either; rapid growth times are hard on a child. Perhaps it’s a small comfort to know that the harder-to-live-with stages do alternate with the calmer times,so parents can count on getting periodic breaks.”
    —Saf Lerman (20th century)

    “Science has fulfilled her function when she has ascertained and enunciated truth.”
    —Thomas Henry Huxley (1825–95)