# Polylog of primitive root of unity

**Calculus**Level 5

\[\large \sum_{\omega\in W} \text{Li}_2(\omega x) = \sum_{n=1}^{2015} a_n \text{Li}_2(x^n)\] The equation above holds true where \(W\) is the set of the \(2015^\text{th}\) primitive roots of unity.

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

**Notation**:

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

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