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The series $G_s(x):=\sum_{n=1}^\infty n^{-s}sin^2(nx)$ is, up to a constant factor, equal to $Li_s(1)-\Re Li_s(e^{2ix})$, where $Li_s(\cdot)$ is a polylogarithm function. $G_2(x)\sim x$ for $x\ll 1$, that is, $\lim_{x\rightarrow 0^+}G_2(x)/x=Const.\neq 0$. $G_3(x)\sim x^2\ln(x)$ for $x\ll 1$. Trivially, $G_s(x)\sim x^2$ if $s>3$.

My question is: is there evidence, that there exists a sequence of factors $\{\theta_n: n\geq 1\}$ such that $\theta_n\geq 0$ and $\sum_{n=1}^\infty \theta_n sin^2(nx)\sim x^\alpha$ for $x\ll 1$ and for choosen (or at least some) power $\alpha\in(1,2)$?

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