Towards a $\mathrm{GL}_n$ variant of the Hoheisel phenomenon

Let $π$ be a unitary cuspidal automorphic representation of $\mathrm{GL}_n$ over a number field, and let $\tildeπ$ be contragredient to $π$. We prove effective upper and lower bounds of the correct order in the short interval prime number theorem for the Rankin-Selberg $L$-function $L(s,π\times\tildeπ)$, extending the work of Hoheisel and Linnik. Along the way, we prove for the first time that $L(s,π\times\widetildeπ)$ has an unconditional standard zero-free region apart from a possible Landau-Siegel zero.

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