The Explicit Sato-Tate Conjecture and Densities Pertaining to Lehmer-Type Questions

TL;DR

This paper proves an explicit form of the Sato-Tate conjecture for newforms, providing density estimates for primes and integers related to their Fourier coefficients, under certain unproven but widely believed hypotheses.

Contribution

It establishes an explicit Sato-Tate distribution result for newforms assuming analytic properties of symmetric power L-functions, and derives density bounds for non-zero Fourier coefficients.

Findings

Quantitative error bounds for prime angle distributions

Lower bounds for the density of integers with non-zero coefficients

Connection between coefficient non-vanishing and quadratic form representations

Abstract

Let $f(z)=\sum_{n=1}^\infty a(n)q^n\in S^{\text{new}}_ k (\Gamma_0(N))$ 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(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 $3.34$. 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 $a(n)\neq0$. In particular, if $\tau$ is the Ramanujan tau function, then under the aforementioned hypotheses, we prove that \[ \lim_{x\to\infty}\frac{\#\{n\leq x:\tau(n)\neq0\}}{x}>1-1.54\times10^{-13}. \] We also discuss the connection between the density of positive integers $n$ for which $a(n)\neq0$ and the number of representations of $n$ by certain positive-definite, integer-valued quadratic forms.