The explicit Sato-Tate Conjecture and densities pertaining to Lehmer-type questions

Abstract:

Let $f(z)=\sum _{n=1}^\infty a_f(n)q^n\in S^{\text {new}}_k (\Gamma _0(N))$ be a newform with squarefree level $N$ that does not have complex multiplication. For a prime $p$, define $\theta _p\in [0,\pi ]$ to be the angle for which $a_f(p)=2p^{( k -1)/2}\cos \theta _p$. Let $I\subset [0,\pi ]$ be a closed subinterval, and let $d\mu _{ST}=\frac {2}{\pi }\sin ^2\theta d\theta$ be the Sato-Tate measure of $I$. Assuming that the symmetric power $L$-functions of $f$ satisfy certain analytic properties (all of which follow from Langlands functoriality and the Generalized Riemann Hypothesis), we prove that if $x$ is sufficiently large, then \[ \left |\#\{p\leq x:\theta _p\in I\} -\mu _{ST}(I)\int _2^x\frac {dt}{\log t}\right |\ll \frac {x^{3/4}\log (N k x)}{\log x} \] with an implied constant of $3.33$. By letting $I$ be a short interval centered at $\frac {\pi }{2}$ and counting the primes using a smooth cutoff, we compute a lower bound for the density of positive integers $n$ for which $a_f(n)\neq 0$. In particular, if $\tau$ is the Ramanujan tau function, then under the aforementioned hypotheses, we prove that \[ \lim _{x\to \infty }\frac {\#\{n\leq x:\tau (n)\neq 0\}}{x}>1-1.54\times 10^{-13}. \] We also discuss the connection between the density of positive integers $n$ for which $a_f(n)\neq 0$ and the number of representations of $n$ by certain positive-definite, integer-valued quadratic forms.