Polylog of root of unity

Calculus Level 5

k=1nLi2(ωkx)=f(n)Li2(x)+g(n)Li2(xn)\large \sum_{k=1}^n \text{Li}_2(\omega^k x)=f(n)\text{Li}_2(x)+g(n)\text{Li}_2(x^n)

The equation above holds true for rational functions ff and gg and ω\omega is the primitive nthn^\text{th} roots of unity for integer n>1n>1. Find (f+g)(1) (f+g)(1) .

Notation: Lin(a){ \text{Li} }_{ n }(a) denotes the polylogarithm function, Lin(a)=k=1akkn.{ \text{Li} }_{ n }(a)=\displaystyle\sum _{ k=1 }^{ \infty }{ \frac { { a }^{ k } }{ { k }^{ n } } }.

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