Monotone chains of Fourier coefficients of Hecke cusp forms

Abstract

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.