A zero density estimate and fractional imaginary parts of zeros for ${\mathrm{GL}}_2$ $L$-functions

Abstract

We prove an analogue of Selberg's zero density estimate for $\zeta (s)$ that holds for any ${\mathrm{GL}}_2$ $L$-function. We use this estimate to study the distribution of the vector of fractional parts of $\gamma \alpha$, where $\alpha \in {\mathbb{R}}^n$ is fixed and $\gamma$ varies over the imaginary parts of the nontrivial zeros of a ${\mathrm{GL}}_2$ $L$-function.

25 pages

References in corpus (4)

• Effective forms of the Sato--Tate conjecture
• Unnormalized differences between zeros of L-functions
• Zeros of ${\mathrm{GL}}_2$ $L$-functions on the critical line
• Simultaneous distribution of fractional parts of Riemann zeta zeros