Monotone chains of Fourier coefficients of Hecke cusp forms

TL;DR

This paper establishes equidistribution results for Fourier coefficients of Hecke cusp forms, demonstrating positive density of certain inequalities among Ramanujan tau values, advancing understanding of their distribution and sign patterns.

Contribution

It proves new equidistribution theorems for Fourier coefficients of Hecke cusp forms, including positive density results for inequalities and progress towards signed versions, under certain conjectures.

Findings

Positive natural density for inequalities among Ramanujan tau values

At least 1/6 relative upper density for triple inequalities of tau values

Conditional density results for chains of inequalities of Fourier coefficients

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 $\tau$ function and any admissible $k$-tuple of distinct non-negative integers $a_1,\ldots,a_k$ the set $$ \{n \in \mathbb{N} : |\tau(n+a_1)| < \cdots < |\tau(n+a_k)|\} $$ 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 $$ \{n \in \mathbb{N} : \tau(n+a_1) < \tau(n+a_2) < \tau(n+a_3)\} $$ 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 Matom\"{a}ki and Radziwill. Our results rely crucially on several key ingredients: i) a multivariate Erd\H{o}s-Kac type theorem for the function $n \mapsto \log|\tau(n)|$, conditioned on $n$ belonging to the set of non-vanishing of $\tau$, 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 Ter\"{a}v\"{a}inen on the logarithmic Elliott conjecture.