Random ordering in modulus of consecutive Hecke eigenvalues of primitive forms

Abstract

Let τ (·) be the classical Ramanujan τ -function and let k be a positive integer such that τ (n) = 0 for 1 ≤ n ≤ k/2. (This is known to be true for k < 10 23 , and, conjecturally, for all k.) Further, let σ be a permutation of the set {1, ..., k}. We show that there exist infinitely many positive integers m such that |τ (m + σ(1))| < |τ (m + σ(2))| < ... < |τ (m + σ(k))|.We also obtain a similar result for Hecke eigenvalues of primitive forms of square-free level.

Monotone chains of Fourier coefficients of Hecke cusp forms

“…In this case, our problem reduces to another widely studied question, that of counting tuples of values of f with prescribed sign patterns. For generic, unbounded multiplicative functions f , the event f (n + a i ) = f (n + a j ) is rare, and so for the purposes of this paper we will count sets like in (1), where the inequalities are to be replaced by strict ones.…”

“…One of the objectives of the present paper is to study (1) for multiplicative functions determined by the Fourier coefficients of non-CM primitive Hecke cusp forms. To this end, we let f (z) := n≥1 b f (n)e(nz) be a primitive non-CM Hecke eigencusp form 1 weight m, square-free level N and trivial nebentypus, normalized so that b f (1) = 1, and let λ f (n) := b f (n)n −(m−1)/2 .…”

“…A famous conjecture of Lehmer states that τ (n) = 0 for all n ≥ 2. Under this assumption, it is natural to upgrade the statement of (1.2) and conjecture that the natural density (the definition of which is recalled below) of the corresponding set exists and is in fact equal to 1 k! .…”

“…To avoid this difficulty, one can therefore consider the corresponding question for |τ (n)|. As an application of the main theorem in the recent beautiful paper [1] (see Theorem 1.1 there), the authors show that if a 1 , . .…”

“…Remark 1.4. We remark that in [1], a suitable set of n ≤ x is constructed such that ω( i≤k (n + a i )) ≪ k 1, and consequently their number is necessarily sparse, specifically ≪ k x/(log x) k−o (1) . In our case, the integers n ≤ x in the conclusion of Corollary 1.3 satisfy ω(n + a i ) ∼ log log x for all 1 ≤ i ≤ k, which naturally provides a density increase.…”