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 \[ Δ(z) = q\prod_{j=1}^{\infty}(1-q^{j})^{24}\ =\ \sum_{n=1}^{\infty}τ(n) q^n \,,q=e^{2πi z} \] (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer's conjecture asserts that $\tau (n)\ne 0$ for all $n\ge 1$; 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 $x\ge 10^{50}$, then \[ \#\{x < p\leq 2x: τ(p) = 0\} \leq 1.22 \times 10^{-5} \frac{x^{3/4}}{\sqrt{\log x}},\] 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.

16 pages, fixed typographical errors and minor computational details. To be published in the International Journal of Number Theory

References in corpus (2)

• The Explicit Sato-Tate Conjecture and Densities Pertaining to Lehmer-Type Questions
• Elliptic Curve Variants of the Least Quadratic Nonresidue Problem and Linnik's Theorem