The Error Term in the Sato-Tate Conjecture (1407.2656v2)

Abstract: Let $f(z)=\sum_{n=1}^\infty a(n)e^{2\pi i nz}\in S_k^{new}(\Gamma_0(N))$ be a newform of even weight $k\geq2$ that does not have complex multiplication. Then $a(n)\in\mathbb{R}$ for all $n$, so for any prime $p$, there exists $\theta_p\in[0,\pi]$ such that $a(p)=2p^{(k-1)/2}\cos(\theta_p)$. Let $\pi(x)=#{p\leq x}$. For a given subinterval $I\subset[0,\pi]$, the now-proven Sato-Tate Conjecture tells us that as $x\to\infty$, [ #{p\leq x:\theta_p\in I}\sim \mu_{ST}(I)\pi(x),\quad \mu_{ST}(I)=\int_{I} \frac{2}{\pi}\sin^2(\theta)~d\theta. ] Let $\epsilon>0$. Assuming that the symmetric power $L$-functions of $f$ are automorphic, we prove that as $x\to\infty$, [ #{p\leq x:\theta_p\in I}=\mu_{ST}(I)\pi(x)+O\left(\frac{x}{(\log x)^{9/8-\epsilon}}\right), ] where the implied constant is effectively computable and depends only on $k,N,$ and $\epsilon$.