The explicit Sato–Tate conjecture for primes in arithmetic progressions

Abstract

Let [Formula: see text] be Ramanujan’s tau function, defined by the discriminant modular form [Formula: see text] (this is the unique holomorphic normalized cuspidal newform of weight 12 and level 1). Lehmer’s conjecture asserts that [Formula: see text] for all [Formula: see text]; since [Formula: see text] is multiplicative, it suffices to study primes [Formula: see text] for which [Formula: see text] might possibly be zero. Assuming standard conjectures for the twisted symmetric power [Formula: see text]-functions associated to [Formula: see text] (including GRH), we prove that if [Formula: see text], then [Formula: see text] 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.