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

TL;DR

This paper develops an explicit Chebotarev variant of the Brun--Titchmarsh theorem and applies it to obtain the first positive unconditional lower bound on the density of integers with nonzero Ramanujan tau-function values.

Contribution

It introduces an explicit Chebotarev theorem variant and applies it to derive unconditional bounds related to Ramanujan's tau-function, advancing understanding of its distribution.

Findings

Proves an explicit Chebotarev variant of the Brun--Titchmarsh theorem.

Establishes a positive unconditional lower bound for the density of integers with nonzero tau(n).

Quantifies the proportion of integers with nonzero tau(n) as greater than 1 - 1.15 x 10^{-12}.

Abstract

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 $$\lim_{x \to \infty} \frac{\#\{1 \leq n \leq x \mid \tau(n) \neq 0\}}{x} > 1-1.15 \times 10^{-12},$$ 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 $\tau(n) \neq 0$.