SIGN CHANGES OF THE COEFFICIENTS OF AUTOMORPHIC <i>L</i>-FUNCTIONS

Lau, Yuk-Kam; Liu, Jianya; Wu, Jie

doi:10.1142/9789814452458_0009

Cited by 10 publications

(12 citation statements)

References 69 publications

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“…The sign problem drew good attentions (e.g. [11,13,14,16,17]) and the widely used approach, especially for somewhat general situations, is based on the first and second moments. This approach, however, may not be efficient for high rank groups due to the poor order estimate for the Rankin-Selberg L-function which is the key ingredient to evaluate the second moment.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Local Behavior of Arithmetical Functions with Applications to Automorphic $L$-Functions

Lau¹,

Liu

2016

Int Math Res Notices

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We derive a Voronoi-type series approximation for the local weighted mean of an arithmetical function that is associated to Dirichlet series satisfying a functional equation with gamma factors. The series is exploited to study the oscillation frequency with a method of Heath-Brown and Tsang [7]. A by-product is another proof for the well-known result of no element in the Selberg class of degree 0 < d < 1. Our major applications include the sign-change problem of the coefficients of automorphic L-functions for GL m , which improves significantly some results of Liu and Wu [14]. The cases of modular forms of half-integral weight and Siegel eigenforms are also considered.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Local Behavior of Arithmetical Functions with Applications to Automorphic $L$-Functions

Lau¹,

Liu

2016

Int Math Res Notices

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“…This theorem can be thought of as a variant of Strum's result about distinguishing two newforms by their Fourier-coefficients. This result can be compared with the results of Lau-Liu-Wu [19], Kohnen [13], Kowalski-Michel-Vanderkam [16], Ram Murty [25] and Sengupta [32].…”

Section: Introductionmentioning

confidence: 70%

“…where θ(ρ) is as in (19). (2) For any ǫ > 0, x ≥ x 0 (P f,N , ǫ), y ≥ x ψ(ρ)+ǫ and 1 ≤ a ≤ q ≤ x ǫ with (a, q) = 1, we have #{x < n ≤ x + y : (n, N ) = 1, n square-free, n ≡ a(mod q) and f (n) = 0} ≫ P f,N , ǫ y/q, where ψ(ρ) is as in (20).…”

Section: Definition 1 Let Us Assume Thatmentioning

confidence: 99%

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

Gun

Kumar

Paul

2019

Journal of Number Theory

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Let f and g be two distinct newforms which are normalized Hecke eigenforms of weights k1, k2 ≥ 2 and levels N1, N2 ≥ 1 respectively. Also let a f (n) and ag(n) be the n-thFourier-coefficients of f and g respectively. In this article, we investigate the first sign change ofwhere p is a prime number. We further study the nonvanishing of the sequence {a f (n)ag(n)} n∈N and derive bounds for first non-vanishing term in this sequence. We also show, using ideas of Kowalski-Robert-Wu and Murty-Murty, that there exists a set of primes S of natural density one such that for any prime p ∈ S, the sequence {a f (p n )ag(p m )} n,m∈N has no zero elements. This improves a recent work of Kumari and Ram Murty. Finally, using B-free numbers, we investigate simultaneous non-vanishing of coefficients of m-th symmetric power L-functions of non-CM forms in short intervals.2010 Mathematics Subject Classification. 11F30, 11F11.

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“…In this paper, we will investigate sign changes of the sequence {a * f (n 2 )} n 1 . This problem has received much attention [10,7,9,11,2,4]. In particular, denoting by n f the smallest integer n such that (1.1) a * f (n 2 ) < 0.…”

Section: Introductionmentioning

confidence: 99%