论文标题
几乎所有的素数都满足了Atkin-Serre的猜想，并不是极端的
Almost all primes satisfy the Atkin-Serre conjecture and are not extremal
论文作者
论文摘要
令$ f（z）= \ sum_ {n = 1}^{\ infty} a_f（n）e^{2πin z} $是一个非CM Holomorphic Cupsidal Newform的Nebentypus的新形式，甚至是整体级别$ k \ geq 2 $。 deligne的猜想证明表明，对于所有Primes $ p $我们证明了100％的Primes $ p $ $ 2p^{\ frac {k-1} {2}}} \ frac {\ log \ log \ log p} {\ sqrt {\ sqrt {\ log p}}} <| a_f（p）我们的证明为特殊集合的大小提供了有效的上限。下边界表明，Atkin-serre的猜想是100％的素数满足的，上限表明$ | a_f（p）| $在0％的prime中尽可能大（即$ p $对于$ f $）。我们的证明使用了第二作者证明的Sato-Tate猜想的有效形式，该猜想依赖于最近由于牛顿和Thorne造成的$ F $对称权力的汽车的证据。
Let $f(z)=\sum_{n=1}^{\infty} a_f(n)e^{2πi n z}$ be a non-CM holomorphic cupsidal newform of trivial nebentypus and even integral level $k\geq 2$. Deligne's proof of the Weil conjectures shows that $|a_f(p)|\leq 2p^{\frac{k-1}{2}}$ for all primes $p$. We prove for 100% of primes $p$ that $2p^{\frac{k-1}{2}}\frac{\log\log p}{\sqrt{\log p}}<|a_f(p)|<\lfloor 2p^{\frac{k-1}{2}}\rfloor$. Our proof gives an effective upper bound for the size of the exceptional set. The lower bound shows that the Atkin-Serre conjecture is satisfied for 100% of primes, and the upper bound shows that $|a_f(p)|$ is as large as possible (i.e., $p$ is extremal for $f$) for 0% of primes. Our proofs use the effective form of the Sato-Tate conjecture proved by the second author, which relies on the recent proof of the automorphy of the symmetric powers of $f$ due to Newton and Thorne.
