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 $$\underset{x\to \mathrm{\infty }}{\lim }\frac{\mathrm{\#}\{1\le n\le x\mid \tau (n)\mathrm{\ne }0\}}{x}>1-1.15\times {10}^{-12},$$ 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 $\tau (n)\mathrm{\ne }0$.