The Explicit Sato-Tate Conjecture For Primes In Arithmetic Progressions

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.