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 Li_s(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

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