Polylog of primitive root of unity

Calculus Level 5

ωWLi2(ωx)=n=12015anLi2(xn)\large \sum_{\omega\in W} \text{Li}_2(\omega x) = \sum_{n=1}^{2015} a_n \text{Li}_2(x^n) The equation above holds true where WW is the set of the 2015th2015^\text{th} primitive roots of unity.

If the value of n=12015an \displaystyle \sum_{n=1}^{2015} a_n can be expressed as AB -\dfrac AB, where AA and BB are coprime positive integers, find A+BA+B.

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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