Infinitely many zeros of additively twisted $L$-functions on the critical line

Abstract

For $f$ a cuspidal modular form for the group $\Gamma_0(N)$ of integral or half-integral weight, $N$ a multiple of $4$ in case the weight is half-integral, we study the zeros of the $L$-function attached to $f$ twisted by an additive character $e^{2\pi i n \frac{p}{q}}$ with $\frac{p}{q}\in \mathbb{Q}$. We prove that for certain $f$ and $\frac{p}{q}\in \mathbb{Q}$, the additively twisted $L$-function has infinitely many zeros on the critical line. We develop a variant of the Hardy-Littlewood method which uses distributions to prove the result.