The Error Term in the Sato-Tate Conjecture

Abstract

Let $f(z)={\sum }_{n=1}^{\infty }a(n)e^{2\pi inz}\in S_k^{new}({\Gamma }_0(N))$ be a newform of even weight $k\ge 2$ 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 $\pi (x)=\#\{p\le x\}$. For a given subinterval $I\subset [0,\pi ]$, the now-proven Sato-Tate Conjecture tells us that as $x\to \infty$, \[ \#\{p\leq x:θ_p\in I\}\sim μ_{ST}(I)π(x),\quad μ_{ST}(I)=\int_{I} \frac{2}π\sin^2(θ)~dθ. \] Let $\epsilon >0$. 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 $\epsilon$.

9 pages

Cited by in corpus (5)

• Effective forms of the Sato--Tate conjecture
• On the value-distribution of symmetric power L-functions
• On the theory of $M$-functions
• Equidistribution of Kloosterman sums over function fields
• Building bridges between Tate conjectures and arithmetic invariants