Joint IAS/Princeton University Number Theory Seminar

I will discuss recent work with Maksym Radziwill in which we show that for any fixed tempered cuspidal representation $\pi$ of $\mathrm{GL}(4)$ over the rationals, there exist infinitely many primitive characters $\chi$ such that the twisted $L$-function $L(s,\pi\times\chi)$ is non-vanishing at the central point $s=1/2$. I will focus on the proof, which involves a mix of ideas.