Local behavior of arithmetical functions with applications to automorphic L-functions

Yuk-Kam Lau (1) , Jianya Liu (2) , Jie Wu (3, 4)

1 Department of Mathematics
2 Shandong University
3 IECL - Institut Élie Cartan de Lorraine
4 LAMA - Laboratoire Analyse et de Mathématiques Appliquées

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.

Domaines: Théorie des nombres [math.NT]

Yuk-Kam Lau, Jianya Liu, Jie Wu. Local behavior of arithmetical functions with applications to automorphic L-functions. International Mathematics Research Notices, 2016, pp.rnw148. ⟨10.1093/imrn/rnw148⟩. ⟨hal-01279192v2⟩