The Explicit Sato-Tate Conjecture For Primes In Arithmetic Progressions

Abstract

Let $\tau(n)$ be Ramanujan's tau function, defined by the discriminant modular form \[ \Delta(z) = q\prod_{j=1}^{\infty}(1-q^{j})^{24}\ =\ \sum_{n=1}^{\infty}\tau(n) q^n \,,q=e^{2\pi i z} \] (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer's conjecture asserts that $\tau(n)\neq 0$ for all $n\geq 1$; since $\tau(n)$ is multiplicative, it suffices to study primes $p$ for which $\tau(p)$ might possibly be zero. Assuming standard conjectures for the twisted symmetric power $L$-functions associated to $\tau$ (including GRH), we prove that if $x\geq 10^{50}$, then \[ \#\{x < p\leq 2x: \tau(p) = 0\} \leq 1.22 \times 10^{-5} \frac{x^{3/4}}{\sqrt{\log x}},\] a substantial improvement on the implied constant in previous work. To achieve this, under the same hypotheses, we prove an explicit version of the Sato-Tate conjecture for primes in arithmetic progressions.