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 , . . . , 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 , . . . , a k the sethas 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 ∈ 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 nonnegative 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 ≤ 2 as consequences of works by Serre and by Matomäki and Radziwi l l. Our results rely crucially on several key ingredients: i) a multivariate Erdős-Kac type theorem for the function n → log |τ (n)|, conditioned on n belonging to the set of non-vanishing of τ , generalizing work of Luca, Radziwi l l and Shparlinski; ii) the recent breakthrough of Newton and Thorne on the functoriality of symmetric power L-functions for GL(n) for all n ≥ 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.

Divisor-bounded multiplicative functions in short intervals

“…Let φ be a holomorphic cuspidal eigenform without CM for SL 2 (Z), of weight k ≥ 2. In a recent paper [6], Klurman and the author considered a particular instance of this problem in which (f (n)) n corresponds to the sequence of Fourier coefficients of such a (correctly normalised) form φ. Writing φ in its Fourier expansion φ(z) = n≥1 a φ (n)e(nz), Im(z) > 0, normalised so that a φ (1) = 1, it was shown among other things that each of the sets of solutions n, m ∈ N to the inequalities a φ (n) < a φ (n + h), a φ (m + h) < a φ (m) has 1 positive upper density.…”

“…Theorem 1.4 actually allows us to recover (1) for the sequence of Fourier coefficients of any holomorphic cusp form without CM for SL 2 (Z), as it is well-known that these sequences consist of algebraic integers in some number field. Moreover, in contrast to a proof using the work [6], we do not need to use an effective version of the Sato-Tate theorem such as that found in [14], thereby avoiding application of the deep theorems of Newton-Thorne [9] on the functoriality of symmetric power lifts of holomorphic cusp forms. Moreover, in the case of forms with (rational) integer-valued Fourier coefficients, we require nothing more than an estimate of Serre [11] on the sparseness of the set of primes {p : a φ (p) = 0}, as well as the prime number theorem for Rankin-Selberg L-functions (see Section 7).…”

“…Here we use the standard notation e(y) := e 2πiy for y ∈ R 6. Crucially, the admissible moduli q here are allowed to grow rapidly relative to δ.…”