Hoheisel现象的$\mathrm{GL}_n$变体
Towards a $\mathrm{GL}_n$ variant of the Hoheisel phenomenon
摘要：
设$π$是数域上$\mathrm{GL}_n$的酉尖点自守表示，且$\tildeπ$与$π$是反义词.本文证明了Rankin-Selberg $L$ -函数$L(s,π\times\tildeπ)$的短区间素数定理中正确阶的有效下界，推广了Hoheisel和Linnik的工作.一路上，我们证明了第一次，$L(s,π\times\widetildeπ)$有一个无条件的标准零自由区域除了一个可能的朗道-西格尔零。
英文摘要：
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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