论文标题
Hecke Cusp形式的傅立叶系数的单调链
Monotone chains of Fourier coefficients of Hecke cusp forms
论文作者
论文摘要
我们证明了与算术归一化的全体形态hecke cusp的傅立叶系数相关的一般等均分配声明（有条件和无条件），$ f_1，\ ldots，f_k $，没有复杂的乘法，具有相等重量（可能不同的）平方英尺和琐事的nebentypus。作为第一个应用程序，我们表明，对于Ramanujan $τ$函数和任何可接受的$ k $ - $ k $ -tuple，由不同的非负整数$ a_1，\ ldots，a_k $ a_k $ the set $ \ {n \ in \ mathbb {n} n} <\ cdots <|τ（n+a_k）| \} $$具有正自然密度。这一结果改善了Bilu，Deshouillers，Gun和Luca [Compos。数学。 （2018），没有。 11，2441-2461]。其次，我们通过表明$ \ {n \ in \ mathbb {n}：τ（n+a_1）<τ（n+a_2）<τ（n+a_3）\} $具有积极的相对上密度至少$ 1/6 $的$ y _ 3 $ $ $（a_3通常，对于这种长度不平等的链条$ k> 3 $，我们表明，在埃利奥特（Elliott）对乘法功能相关性的猜想的假设下，该集合的相对自然密度为$ 1/k！。我们的结果至关重要地依赖于几种关键成分：i）函数$ n \ mapsto \ log | f |τ（n）| $的多变量erdős-kac类型定理，以$ n $为条件，属于$ n $属于$τ$的$ n $，是$τ$的集合，luca，radziwill and radziwill and shparparinski的普遍化工作， ii）牛顿和索恩最近在所有$ n \ geq 2 $的$ \ text {gl}（n）$的对称功能$ l $ functions的功能上的突破，并将其应用于sato-tate cosivenure的定量形式； iii）陶和特雷宁在对数埃利奥特猜想上的工作。
We prove general equidistribution statements (both conditional and unconditional) relating to the Fourier coefficients of arithmetically normalized holomorphic Hecke cusp forms $f_1,\ldots,f_k$ without complex multiplication, of equal weight, (possibly different) squarefree level and trivial nebentypus. As a first application, we show that for the Ramanujan $τ$ function and any admissible $k$-tuple of distinct non-negative integers $a_1,\ldots,a_k$ the set $$ \{n \in \mathbb{N} : |τ(n+a_1)| < \cdots < |τ(n+a_k)|\} $$ has positive natural density. This result improves upon recent work of Bilu, Deshouillers, Gun and Luca [Compos. Math. (2018), no. 11, 2441-2461]. Secondly, we make progress towards understanding the signed version by showing that $$ \{n \in \mathbb{N} : τ(n+a_1) < τ(n+a_2) < τ(n+a_3)\} $$ has positive relative upper density at least $1/6$ for any admissible triple of distinct non-negative integers $(a_1,a_2,a_3).$ More generally, for such chains of inequalities of length $k > 3$ we show that under the assumption of Elliott's conjecture on correlations of multiplicative functions, the relative natural density of this set is $1/k!.$ Previously results of such type were known for $k\le 2$ as consequences of works by Serre and by Matomäki and Radziwill. Our results rely crucially on several key ingredients: i) a multivariate Erdős-Kac type theorem for the function $n \mapsto \log|τ(n)|$, conditioned on $n$ belonging to the set of non-vanishing of $τ$, generalizing work of Luca, Radziwill and Shparlinski; ii) the recent breakthrough of Newton and Thorne on the functoriality of symmetric power $L$-functions for $\text{GL}(n)$ for all $n \geq 2$ and its application to quantitative forms of the Sato-Tate conjecture; and iii) the work of Tao and Teräväinen on the logarithmic Elliott conjecture.
