A Selberg-type zero-density result for twisted $\rm GL_2$ $L$-functions and its application

TL;DR

This work establishes an unconditional Selberg-type zero-density bound for twisted GL_2 L-functions $L(s,f\otimes\chi)$ as $\chi$ ranges over primitive characters modulo a prime $q$, yielding a decay in the density of zeros off the critical line. Using a mollified second-moment framework, the authors derive an asymptotic formula for the even moments of the argument $S(t,f\otimes\chi)$ and prove a central limit theorem for its distribution as $q\to\infty$. The approach combines mollification, zero-density arguments, and a careful decomposition of $S$ into main-principal convergent parts and residual terms, providing unconditional results in a GL_2 twist context. These findings illuminate the vertical distribution of zeros and the statistical behavior of $\arg L(\tfrac12+i t,f\otimes\chi)$ across Dirichlet twists, with implications for equidistribution phenomena and analytic estimates in the twisted automorphic setting.

Abstract

Let $f$ be a fixed holomorphic primitive cusp form of even weight $k$, level $r$ and trivial nebentypus $χ_r$. Let $q$ be an odd prime with $(q,r)=1$ and let $χ$ be a primitive Dirichlet character modulus $q$ with $χ\neqχ_r$. In this paper, we prove an unconditional Selberg-type zero-density estimate for the family of twisted $L$-functions $L(s, f \otimes χ)$ in the critical strip. As an application, we establish an asymptotic formula for the even moments of the argument function $S(t, f \otimes χ)=π^{-1}\arg L(1/2+ıt, f\otimesχ)$ and prove a central limit theorem for its distribution over $χ$ of modulus $q$.

Theorems & Definitions (27)

• Theorem 1.1
• Theorem 1.2
• Remark 1
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Lemma 2.4