Multiplication Theorem - Polylogarithm

Polylogarithm

The duplication formula takes the form

The general multiplication formula is in the form of a Gauss sum or discrete Fourier transform:

$k^{1-s} \operatorname{Li}_s(z^k) = \sum_{n=0}^{k-1}\operatorname{Li}_s\left(ze^{i2\pi n/k}\right).$

These identities follow from that on the periodic zeta function, taking z = log q.

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