Modular forms and an explicit Chebotarev variant of the Brun–Titchmarsh theorem

We prove an explicit Chebotarev variant of the Brun–Titchmarsh theorem. This leads to explicit versions of the best known unconditional upper bounds toward conjectures of Lang and Trotter for the coefficients of holomorphic cuspidal newforms. In particular, we prove that

$$
\lim_{x\to\infty} \frac{\left\{1\leq n \leq x : \tau (n) \neq 0 \right\}}{x} > 1 - 1.15\times 10^{-12}
$$

where $\tau(n)$ is Ramanujan’s tau-function. This is the first known positive unconditional lower bound for the proportion of positive integers n such that $\tau (n) \neq 0$. This is joint work with Daniel Hu and Alexander Shashkov.