Simultaneous non-vanishing and sign changes of Fourier coefficients of modular forms

Abstract

Let f and g be two Hecke-Maass cusp forms of weight zero for SL 2 (Z) with Laplacian eigenvalues 1 4 +u 2 and 1 4 +v 2 , respectively. Then both have real Fourier coefficients say, λ f (n) and λ g (n), and we may normalize f and g so that λ f (1) = 1 = λ g (1). In this article, we first prove that the sequence {λ f (n)λ g (n)} n∈N has infinitely many sign changes. Then we derive a bound for the first negative coefficient for the same sequence in terms of the Laplacian eigenvalues of f and g.

The first simultaneous sign change and non-vanishing of Hecke eigenvalues of newforms

“…Theorem 3 shows that for all primes p with (p, N 1 N 2 ) = 1, the non-zero elements of the sequence A p has positive density and hence does not follow from Theorem 3 of [9]. Our next theorem strengthens Theorem 1.2 of Kumari and Ram Murty [18].…”

“…In this article, we will study simultaneous non-vanishing of Hecke eigenvalues using Bfree numbers. This question was first considered by Kumari and Ram Murty [18]. We now introduce the set of B-free numbers as constructed by Kowalski, Robert and Wu [17].…”

“…where ρ ∈ [0, 1] and η ρ 's are real numbers with η 1 > 1. Let us define (18) B P := P ∪ {p 2 | p ∈ P − P}.…”

“…Theorem 16 (Wu and Zhai). Let B P be as in (18). For any ǫ > 0, x ≥ x 0 (P, ǫ), y ≥ x ψ(ρ)+ǫ and 1 ≤ a ≤ q ≤ x ǫ with (a, q) = 1, one has #{x < n ≤ x + y : n is B P -free and n ≡ a(mod q)} ≫ P,ǫ y/q, if 166 173 < ρ ≤ 1.…”

“…Next we investigate sign changes of the sequence {a f (n)a g (n 2 )} n∈N in short intervals. This question of sign change for the sequence {a f (n)a g (n)} n∈N in short intervals was considered by Kumari and Ram Murty (see [18,Theorem 1.6]). Here we prove the following.…”