Paper detail

The Error Term in the Sato-Tate Conjecture

Let $f(z)=\sum_{n=1}^\infty a(n)e^{2πi nz}\in S_k^{new}(Γ_0(N))$ 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 $θ_p\in[0,π]$ such that $a(p)=2p^{(k-1)/2}\cos(θ_p)$. Let $π(x)=\#\{p\leq x\}$. For a given subinterval $I\subset[0,π]$, the now-proven Sato-Tate Conjecture tells us that as $x\to\infty$, \[ \#\{p\leq x:θ_p\in I\}\sim μ_{ST}(I)π(x),\quad μ_{ST}(I)=\int_{I} \frac{2}π\sin^2(θ)~dθ. \] Let $ε>0$. Assuming that the symmetric power $L$-functions of $f$ are automorphic, we prove that as $x\to\infty$, \[ \#\{p\leq x:θ_p\in I\}=μ_{ST}(I)π(x)+O\left(\frac{x}{(\log x)^{9/8-ε}}\right), \] where the implied constant is effectively computable and depends only on $k,N,$ and $ε$.