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

Abstract

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.

On Signs of Fourier Coefficients of Hecke-Maass Cusp Forms on $\mathrm{GL}_3$

“…The first part improves a result of Lau, Liu, and Wu [18] who obtained ≫ X 1/3 sign changes (which in turn improved an earlier result of Meher and Murty [35]). We also note that more generally sign changes of the sequence {ℜA(m, 1)} m≤X have been considered by Hulse, Kuan, Lowry-Duda, and Walker [10] who obtained ≫ ε X 2/23−ε sign changes in this case.…”

“…In the second step we use (18) for forms satisfying the Ramanujan-Petersson conjecture at prime p, but the treatment of the rest of the forms requires additional explanation. Let us define the polynomial f :…”

“…The proof of Theorem 1 follows a different path compared to the method used in [18]. Our strategy is to show that short interval [x, x + H] contains a sign change for many x ∼ X (here and subsequently x ∼ X means X ≤ x ≤ 2X) with H as small as possible whereas Lau, Liu, and Wu proceeded by using a Voronoi type series approximation for the weighted mean of the Fourier coefficients together with the method of Heath-Brown and Tsang [9].…”

A large sieve inequality of Elliott–Montgomery–Vaughan type for automorphic forms on GL$_3$

“…There are various results in the GL 2 case (see [6], [7], [14], [15], etc). When it comes to GL n , the automorphic form is usually supposed to be self-dual, so that it has real Fourier coefficients (see [8], [11], [12], [18], for example).…”