论文标题
Brun-titchmarsh定理的模块化形式和明确的Chebotarev变体
Modular forms and an explicit Chebotarev variant of the Brun-Titchmarsh theorem
论文作者
论文摘要
我们证明了Brun-titchmarsh定理的明确Chebotarev变体。这导致了最著名的无条件上限的明确版本，朝向lang and Trotter的猜想，以构成全体形态尖锐的新形式的系数。特别是，我们证明了$ \ lim_ {x \ to \ infty} \ frac {\＃\ {1 \ leq n \ leq n \ leq x \ mid th \ midτ（n）\ neq 0 \ neq 0 \}} {x} {x}> 1-1.15 \ times 10^times 10^{ - 12^{ - 12} { - 12}，$ ram $ ram pran $ taujany是$ tauj（这是正整数$ n $比例的第一个已知的正无条件下限，因此$τ（n）\ neq 0 $。
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 $$\lim_{x \to \infty} \frac{\#\{1 \leq n \leq x \mid τ(n) \neq 0\}}{x} > 1-1.15 \times 10^{-12},$$ where $τ(n)$ is Ramanujan's tau-function. This is the first known positive unconditional lower bound for the proportion of positive integers $n$ such that $τ(n) \neq 0$.
